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23  WEST  MAIN  STREET 

WEBSTER,  N.Y.  )4S80 

(716)  872-4503 


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CIHM/ICMH 

Microfiche 

Series. 


CIHM/ICMH 
Collection  de 
microfiches. 


Canadian  Institute  for  Historical  Microreproductions  /  Institut  Canadian  de  microreproductions  historiques 


Technical  and  Bibliographic  Notes/Notes  techniques  et  bibliographiques 


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D 


n 


• 


D 


Coloured  covers/ 
Couverture  de  couleur 


I      I    Covers  damaged/ 


Couverture  endommagee 

Covers  restored  and/or  laminated/ 
Couverture  restaurde  et/ou  pelliculde 

Cover  title  missing/ 

Le  titre  de  couverture  manque 

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Encre  de  couleur  (i.e.  autre  que  bleue  ou  noire) 


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modification  dans  la  m^thode  normale  de  filmage 
sont  indiquds  ci-dessous. 


□ 

D 
D 


v/ 


n 


V 


G 

n 

n 


Coloured  pages/ 
Pages  de  couleur 

Pages  damaged/ 
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Pages  restored  and/or  laminated/ 
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This  item  is  filmed  at  the  reduction  ratio  checked  below/ 

Ce  document  est  film6  au  taux  de  reduction  indiqud  ci-dessous. 


10X 

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y 

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32X 


re 

letails 
3s  du 
nodifier 
3r  une 
ilmage 


The  copy  filmed  here  has  been  reproduced  thanks 
to  the  generosity  of: 

Ralph  Pickard  Bell  Library 

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g6n6rosit6  de: 

Ralph  Pickard  Bell  Library 

Mount  Allison  University 


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plus  grand  soin,  compte  tenu  de  la  condition  et 
de  la  nettet6  de  l'exemplaire  film6,  et  ^n 
conformity  avec  les  conditions  du  contrat  de 
filmage. 


es 


Original  copies  in  printed  paper  covers  are  filmed 
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or  illustrated  impression. 


The  last  recorded  frame  on  each  microfiche 
shall  contain  the  symbol  — ^  (meaning  "CON- 
TINUED "),  or  the  symbol  V  (meaning  "END"), 
whichever  applies. 


Les  exemplaires  originaux  dont  la  couverture  en 
papier  est  imprimde  sont  fi!m6s  en  commengant 
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dernidre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration,  soit  par  le  second 
plat,  selon  le  cas.  Tous  les  autres  exemplaires 
originaux  sont  film6s  en  commengant  par  la 
premidre  page  qui  comporte  une  empreinte 
d'impression  ou  d'illustration  et  en  terminant  par 
la  dernidre  page  qui  comporte  une  telle 
empreinte. 

Un  des  symboles  suivants  apparaitra  sur  la 
dernidre  image  de  cheque  microfiche,  selon  le 
cas:  le  symbole  — ^  signifie  "A  SUIVRE",  le 
symbole  V  signifie  "FIN". 


Maps,  plates,  charts,  etc.,  may  be  filmed  at 
different  reduction  ratios.  Those  too  large  to  be 
entirely  included  in  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
method: 


Les  cartes,  planches,  tableaux,  etc.,  peuvent  Stre 
film^s  d  des  taux  de  reduction  diff^rents. 
Lorsque  le  document  est  trop  grand  pour  etre 
reproduit  en  un  seul  cliche,  il  est  film6  d  partir 
de  Tangle  sup^rieur  gauche,  de  gauche  d  droite, 
et  de  haut  en  bas,  en  prenant  le  nombre 
d'images  n^cessaire.  Les  diagrammes  suivants 
illustrent  la  m^thode. 


errata 
i  to 


9  pelure, 
on  d 


n 


32X 


1 

2 

3 

1 

2 

3 

4 

5 

6 

NE  WCOMB'  S 

Mathematical  Course. 

* 

I.    SCHOOL  COURSE. 

Algebra  for  Schools, $1.20 

Key  to  Algebra  for  Schools,        ....  1.20 

Plane  Geometry  and  Trigonometry,  with  Tables,  1.40 

The  Essentials  of  Trigonometry,   ....  1.25 

II.    COLLEGE   COURSE. 

Algebra  for  Colleges, $1.60 

Key  to  Algebra  for  Colleges,       ....  1.60 

Elements  of  Geometry, 1.50 

Plane  and  Spherical  Trigonometry,  with  Tables,  2.00 

Trigonometry  {separate), 1.50 

Tables  (i^eparate)^ 1.40 

Elements  of  Analytic  Geometry,      .        .        .  1.50 

Calculus  (in  preparation), 

Astronomy  (Newcomb  and  Holden,)              .         .  2.50 

The  same,  Briefer  Course,          ....  1.40 

HE/\/RY  HOLT  &  CO.,  Publishers.  New  York. 


S' 


CV///^"    V^t:? 


^r.i 


NEWVOMB'S  MATHEMATICAL    SERIES. 


A  L  ( r  E  B  \l  A 


^  .      '(.    .     JH-  \\\.\  I  » lA. 


FOR 


SCHOOLS     AND     COLLEGES 


BY 


siMOis^  ne\vt:omp> 

P)-ofeKsor  of  Mntfiemafirs,    Vtntcd   States  Aacy 
FIFTH  EDITION,  REVISED, 


HENRY 


CovvRiouT,  18«1,  1884. 

BY 

Henry  Holt  &  Co. 


kn 


Thi 

that  ] 
and  c 
impro 
feach( 
prepa 
adinis 


THOWS 
HT.NG  AND    BOOKBINOmC.  rOMPANV, 
NEW    YORK. 


PREFACE. 


Toe  course  of  algebra  cmbodiod  in  the  present  work  is  substantially 
that  pursued  by  students  in  our  best  preparatory  and  scientitie  schools 
and  cplleges,  with  such  extensions  as  seemed  necessary  to  afford  an 
improved  basis  for  more  advanced  studies.  For  the  convenience  of 
teachers  the  work  is  divided  into  two  parts,  the  first  adapted  to  well- 
[irepared  beginners  and  coniprising  about  what  is  commonly  required  for 
admission  to  coll(,'ge ;  and  the  second  designed  for  tlu;  more  advanced 
fjenoral  studimt.  As  the  work  deviates  in  several  points  from  the  models 
most  familiar  to  our  teachers,  a  statement  of  the  principles  on  which  it  is 
constructed  may  be  deemed  appropriate. 

One  well-known  principle  underlying  the  acquisition  of  knowledge  is 

tiiat  an  idea  cannot  be  fully  grasped  by  the  youthful  mind  unless  it  is 

3  presented  under  a  concrete  form.      Whenever  possible  an  abstract  idea 

i  iimst  be  embodied  in  some  visible  representation,  and  all  general  theorems 

I  nuist  be  presented  in  a  variety  of  special  forms  in  wliich  they  may  be 

I  seen   inductively.     In  accordance  with  this  principle,   numerical  exam- 

]  pies  of  nearly  all  algebraic  operations  and  theorems  iiave  been  presented. 

:■  For  the  purpose  of  illustration,  numbers  have  been  preferred  to  literal 

I  symbols  when  they  would  serve  the  purpose  equally  well.     The  relations 

I  of  positive  and  negative  algebraic  quantities  have  been  represented  by 

f  lines  and  directions  from  the  beginning  in  order  that  the  pupil  might  bo 

I  able  to  give,  not  only  a  numerical,  but  a  visible,  meaning  to  all  algebraic 

I  quantities.     Should  it  appear  to  any  one  that  we  thus  detract  from  the 

jgenerality  of  algebraic  quantities,  it  is  sufficient  to  reply  that  the  system 

jis  the  same  which  mathematicians  use  to  assist  their   conceptions  of 

ulvanced  algebra,  and  without  which  they  would  never  have  been  able 

to  grasp  the  complicated  relations  of  imaginary  quantities.     Algebraic 


^ 


IV 


riiUFACK 


operations  with  pure  numbers  .-iro  made  to  precede  the  use  of  symbols, 
and  the  latter  are  introduced  only  after  the  pupil  lias  had  a  certain 
amount  of  familiarity  with  the  distinction  between  al^t'jebraic  and  numer- 
ical operations. 

Another,  but,  uiifoitunat<;ly,  a  li^ss  familiar  fact  is,  that  all  mathematical 
citiicc|iii()ii.s  icijiiin;  tiiau  to  become  engrafted  iip(jn  the  mind,  and  Oic 
more  tiiiK!  tin;  greater  tlieir  ab.strusenesa.  It  is,  the  juithor  conceives, 
from  a  failure  to  take  account  of  this  fact,  rather  tliuii  from  any  inherent 
defect  in  the  minds  of  our  youtli,  that  we  are  to  attrilnite  the  backward 
state  of  mathematical  instruction  in  this  country,  as  compared  with  the 
continent  of  Europe.  Let  us  take  for  instance  the  case  of  the  student 
commencing  tlie  calculus.  On  the  system  Avhich  was  almost  universal 
among  us  a  few  years  ago,  and  which  is  still  widely  prevalent,  he  is  con- 
fronted at  the  outset  with  a  number  of  entirely  n(nv  conceptions,  such 
as  those  of  variables,  functions,  increments,  infinitesimals  and  limits. 
In  his  first  lesson  he  finds  these  all  combined  with  a  notation  so  entirely 
(litTerent  from  that  to  which  he  has  been  accustomed,  that  belbre  the 
new  ideas  and  forms  of  thought  can  take  permanent  root  in  his  mind, 
he  is  through  with  the  subject,  and  all  that  he  has  learned  is  apt  to  vanish 
from  his  memory  in  a  few  months. 

The  author  conceives  that  the  true  method  of  meeting  this  difficulty  is 
to  adopt  the  French  and  Grerman  plan  of  teaching  algebra  in  a  broader 
way,  and  of  introducing  the  more  advanced  conceptions  at  the  earliest 
practicable  period  in  the  course.  Accordingly,  the  attempt  is  made  in  the 
present  work  to  introduce  each  advanced  conception,  disguised  perhaps 
under  some  simple  form,  in  advance  of  its  general  enunciation  and  at  as 
early  a  period  as  the  student  can  be  expected  to  understand  it.  In  doing 
this,  logical  order  is  frequently  sacrificed  to  the  exigencies  of  the  case, 
because  there  are  several  subjects  with  which  a  certain  amount  of  famil- 
iarity must  be  acquired  before  the  pupil  can  even  clearly  comprehend 
general  statements  respecting  them. 

A  third  feature  of  the  work  is  that  of  subdividing  each  subject  as 
minutely  as  possible,  and  exercising  the  pupil  on  the  details  preparatory  to 
combining  them  into  a  whole.  To  cite  one  or  two  instances :  a  difficulty 
which  not  only  the  beginner  but  the  expert  mathematician  frequently 
meets  is  that  of  stating  his  conceptions  in  algebraic  language.  Exercises 
in  such  statements  have  therefore  been  made  to  precede  any  solution  of 


^ 


I 


phi: FACE. 


'  ol'  synibob, 
ad  a  certain 
;  and  nunicr- 

MiaUienuitioal 
nind,  and  'he 
lor  conceives. 

any  inliercnt 
the  backward 
ared  with  the 
)r  the  student 
lost  universal 
■nt,  he  is  con- 

eptions,  such 
s  and  limits, 
on  so  entirely 
lat  belbre  the 
t  in  his  mind, 
5  apt  to  vanish 

his  difficulty  is 
a  in  a  broader 
at  the  earliest 
is  made  in  the 
r^uised  perhaps 
ition  and  at  as 
.1  it.  In  doing 
les  of  the  case, 
lount  of  famil- 
ly  comprehend 

ach  subject  as 
preparatory  to 
es:  a  difficulty 
cian  frequently 
ige.  Exercises 
my  solution  of 


i 


problems.  In  general  eacli  i)rincipl»'  whidi  in  to  be  presented  or  used  Ls 
ptntt'd  Hingly,  and  the  pupil  i.s  practiced  upon  it  before  })rocoeding  to 
another. 

Subjects  liave  for  the  most  part  been  omitted  whirli  do  not  find  appli- 
cation either  in  the  work  itself  or  in  Hubscquent  parts  of  the  usual  courbo 
of  mathematics,  or  which  do  not  cmulure  to  u  mathematical  training. 
Sturm's  Theorem  has  been  entirely  omitted,  and  a  more  simple  process 
substituted.  The  subject  of  the  greatest  common  divisor  of  two  polyno- 
;  ,ials  has  been  i)osti)nned  to  what  the  author  considers  its  proper  jdace, 
in  the  genwral  theory  of  equations.  It  has,  however,  been  presented  in 
Biich  a  form  that  it  can  be  taught  to  pupils  preparing  for  colleges  where 
it  is  still  required  for  admission. 

Thoroughness  at  each  step  has  been  aimed  at  rather  than  multiplicity 
of  subjects.  It  is,  the  autlior  conceives,  a  great  and  too  common 
mistake  to  present  a  mathematical  subject  to  the  mind  of  the  student 
wiUiout  sufficient  lulnoss  of  explanation  and  variety  of  illustration  to 
enable'  iiim  to  comprehend  and  api>ly  it.  If  ho  has  not  time  to  master  a 
complete  course,  it  is  better  to  omit  entirely  what  is  least  necessary  than 
to  gain  time  by  going  rapidly  over  a  great  number  of  things.  Some 
liints  to  those  who  may  not  have  time  to  master  the  whole  work  may 
therefore  be  acceptable. 

Part  I  is  essential  to  every  one  desiring  to  make  use  of  algebra.  Book 
VIII,  especially  the  concluding  sections  on  notation,  is  to  be  thoroughly 
mastered,  before  going  farther,  as  forming  the  foundation  of  advanced 
algebra ;  and  affording  a  very  easy  and  valuable  discipline  in  the  language 
of  mathematics.  Afterward,  a  selection  may  be  made  according  to  cir- 
cumstances. The  student  who  is  pursuing  the  subject  for  the  sole 
purpose  of  liberal  training,  and  without  intending  to  advance  beyond  it, 
will  find  the  theories  of  numbers  and  the  combinatory  analysis  most 
worthy  of  study.  The  theory  of  probf-bilities  and  the  method  in  which 
it  is  applied  to  such  practical  questions  as  those  connected  with  insurance 
will  be  of  especial  value  in  training  his  judgment  to  the  affairs  of  life. 

The  student  who  mtends  to  take  a  full  course  of  mathematics  with  a 
view  of  its  application  to  physics,  engineering,  or  other  subjects,  may,  if 
necessary,  omit  the  book  on  the  theory  of  numbers,  and  portions  of  the 
chapter  on  the  summation  of  series.  Functions  and  the  functional  notation, 
the  doctrine  of  Umits,  and  the  general  theory  of  equations  will  claim  his 


vi 


riih'FACK 


I 


cspocial  ationtion,  whilo  tho  tluory  of  irnaf,'inftry  qiinntltioa  will  bo  studied 
uiaiiily  to  hocuio  tlior<)ui,'linos.s  "n  siibse(iuonf,  parts  of  lii^  ciourse. 

As  it  luw  fruquoiitly  bfcii  a  pi'Xt  of  the  author's  duty  to  ascortain  wliat 
is  n.-ally  left  of  a  course  of  matlieinatical  study  in  tlio  minds  of  tlioso 
who  have  been  throui^ii  colleg(3,  sonio  hints  on  thu  l)t'st  methods  of 
study  in  connection  witli  tlic  present  worii  may  be  excused.  If  asked 
to  point  out  tlie  j^reutest  error  in  our  usual  system  of  mathematical 
instruction  from  the  common  school  upward,  he  would  reply  that  it  con- 
sisted in  expending  too  much  of  the  mental  power  of  tho  student  upon 
problems  and  exeicises  above  his  capacity.  With  the  exception  of  the 
fundamental  routine-operations,  problems  and  exercises  should  be  confined 
to  insuring  a  proper  understanding  of  the  principles  involved :  this  onco 
ascertained,  it  is  better  that  the  student  should  go  on  rather  than  expend 
time  in  doing  what  it  is  certain  he  can  do.  Problems  of  some  difficulty 
are  found  among  the  exercises  of  the  present  work ;  they  are  inserted 
rather  to  give  the  teacher  a  good  choice  from  which  to  select  than  to 
require  that  any  student  should  do  them  all. 

It  would,  the  author  conceives,  be  found  an  improvement  on  our  usual 
system  of  teaching  algebra  and  geometry  successively  if  the  analytic  and 
the  geometric  courses  of  mathematics  were  pursued  simultaneously.  Tho 
former  would  include  algebra  and  the  calculus,  the  latter  elementary 
geometry,  trigonometry,  and  analytic  geometry.  The  analytic  course 
would  then  furnish  methods  for  the  geometric  one,  and  the  latter  would 
furnish  applications  and  illustrations  for  the  analytic  one. 

The  Key  to  the  work  contains  not  only  the  usual  solutions,  but  tho 
explanations  and  demorstrations  of  the  less  familiar  theorems,  and  a 
number  of  additional  problems. 

The  author  desires,  in  conclusion,  to  express  his  obligation  to  the  many 
friends  who  have  given  him  suggestions  respecting  the  work,  and  espe- 
cially to  Professor  J.  Howard  Gore,  of  the  Columbian  University,  who 
has  furnished  solutions  to  most  of  the  problems,  and  given  the  benefit  of 
his  experience  on  many  points  of  detail. 


Note. — Answers  to  exercises,  rcquiriinj  ralculafion  or  icritten  irork,  ar<? 
published  separately  in  pamphlet  form,  and  will  be  supplied  without 
charge  when  applied  for  by  teachers. 


1 


11  be  studied 
rse. 

jortiiin  what 
ids  of  tliose 
motliods  of 
1.  If  asked 
iiatheiiiatieal 
r  that  it  eon- 
;tudent  upon 
ption  of  the 
1  be  confined 
d :  tills  onco 
than  expend 
me  uiffieulty 
are  inserted 
3lect  than  to 

on  our  usual 

analytic  and 

ously.     The 

elementary 

ytic  course 

atter  would 

ns,  but  the 
urns,   and  a 

bo  the  many 
,  and  esi»«;- 
orsity,  who 
ic  benefit  of 


n  iroric,  are 
ed   without 


^ 


I 


1 


TABLE    OF    CONTEXTS. 


PART     I. —ELEMENTARY     COURSE. 

HOOK   I.— THE   AUJEBRAIC   LAN(UTAGE. 

I'llAPTKU     I.— Al,(iKnilAIC     NUMBEKS     AND     Ol'EHATlONS,     ',].        (Jcncfal 

Dcnnitions,   U.     Algebraic   Numbers,  4.      Algebraic  Addition,  G. 
Subtraction,  8.    Multiplication,!).     Divibiou,  11. 

CiiAPTEU  II.— Algebraic  Symbols,  13.  Symbols  of  Quantity,  12. 
Signs  of  Operation,  13. 

CiiiU'TER  III.— Formation  op  Compound  Expressions,  17.  Funda- 
mental Principles,  17.    Definitions,  11). 

Chapter  IV. — Construction  of  Algebraic  Expressions,  23.  Exer- 
cises in  Algebraic  Language,  25. 

BOOK   II.— ALGEBRAIC  OPERATIONS. 

General  Remarks,  28.     Definitions,  28. 

Chapter  I. — Algebraic  Addition  and  Subtraction,  30.  Algebraic 
Addition,  30.  Algebraic  Subtraction,  33.  Clearing  of  Parenthe- 
ses, 35.    Compound  Parentheses,  37. 

Chapter  II. — Multiplication,  38.  General  Laws  of  Multiplication,  38. 
Multiplication  of  Positive  Monomials,  40.  Rule  of  Signs  in 
Multiplication,  41.  Products  of  Polynomials  by  Monomials,  44 
Multiplication  of  Polynomials  by  Polynomials,  47. 

Chapter  III. — Division,  53.  Division  of  Monomials  by  Monomials,  53. 
Rule  of  Signs  in  Divisicm,  53.     Division  of  Polynomials  by  Mono- 


VIII  VONTKNTS. 

mlals,  r»|.  Factors  nnd  MuItiplrB,  55.  FftctorH  of  IMnf)inialH,  W, 
LfiiHt  Coimnoii  Miiltiplt",  01.  Division  of  one  Polynomial  by 
niiotluT,  (i'J. 

CiLM'TKii  IV.— Of  Ai.okuuaic  Fkactions,  07.  Nc^nitivc  ExiMnicntH,  71. 
Diwsi'ction  of  Fraotionti,  7J1.  Ay^^rogution  of  FractioiiH,  71.  I'-actor- 
in^  FractionH,  78.  Miiltiplicution  and  DiviHion  of  Fractious,  IW. 
Division  of  one  Fniction  l>y  atuHluT,  82.  Kcciprocal  Hoiations  of 
Multiplication  and  Division,  H;J.  ' 


1 


.<? 


BOOK    III.-OF  EQUATIONS. 

CuAPTKn  I  — Tfik  Reduction  of  E(iUATioNw,  85.  Axioms,  87.  Opera- 
tions of  Addition  and  Subtraction — transposing  'IVrnis,  87. 
Oj)cration  of  Multiplication,  89.  Reduction  to  the  Nonnal  Form, 
90.     Degree  of  Ecjuations,  9J3, 

CnAPTEii  II. — Equations  of  the  First  DKonEf';  with  One  Un- 
known Quantity,  94.  Problems  leading  to  Simple  Equations,  99. 
Problem  of  the  Couriers,  105.    ProbleniH  of  ('ircular  Motion,  108. 

Chapter  III.— Equations  of  the  First  Degree  with  Several 
Unknown  Quantities,  109.  Equations  with  Two  Unknown 
Quantities,  109.  Solution  of  a  pair  of  Simultaneous  Equations 
containing  Two  Unknown  Quantities,  109,  Elimination  by  Com- 
parison, 110.  Elimination  by  Substitution,  111.  Elimination  by 
Addition  or  Subtraction,  112.  Problem  of  the  Sum  and  DiflTor- 
ences,  113.  Equations  of  the  First  Degree  with  Three  or  More 
Unknown  Quantities,  116.  Elimination,  116.  Equivalent  and 
Inconsistent  Equations,  121. 

Chapter  IV. — Of  Inequalities,  123. 


:/; 


BOOK   IV.— RATIO  AJ^D  PROPORTION. 
Chapter  I.— Nature  of  a  Ratio,  128.    Properties  of  Ratios,  132. 

Chapter  II. — Proportion,  133.     Theorems  of  Proportion,  134.    Tlie 
Mean  Proportional,  138.     Multiple  Proportions,  139. 


I 


iiir)inialfl,  r»S. 
•lynMrnial    liy 

x|K)ii('nt8,  71. 

,  74.    Factor- 

Kractioim,  ?.>. 

HuIatioiiH  of 


9,  87.    Opera- 
Terms,    87. 
onnal  Form, 

n  One   Un- 

j^quations,  99. 
lotion,  108. 

ni  Several 
vo  Unknown 
IS  Equations 
ion  by  Com- 
imination  by 
11  and  Diffor- 
iree  or  More 
uivalent  and 


coyThwm  IX 


BOOK   v.— OF   I'OWFltS    .WD   R()OTft. 

Cn.\i*TEii  I. — iNVfi  ,t'7i()N,  IM.  Involution  of  ProduotH  and  (^uotientH, 
114.  Involution  of  Powrrn,  145.  Cane  of  Ni'ifativc  KxponcntH,  147. 
Alj^fbrair  Si^'u  of  Powers,  148.  Involution  <»f  Hinoinialh — the 
IJinoniial  Theorem,  1 18.     S<iuare  of  a  Polynomial,  l.'jiJ. 

CriAPTER  II.— Evolution  and  Fuactfonal  Exponents,  155.  Powers 
of  Exprec«ions  with  P'nu-tional  Exponents,  157. 

Cuaptkk  III.— Reduction  of  Ihuatujnal  Exprehhions,  159.  Defini- 
tions, 159.  Ag^^regation  of  Similar  Terms,  HiO.  Factoring  Surds, 
101.  Perfect  S<juarea,  100  To  Complete  the  Square,  107.  Irra- 
tional  Factors,  109. 


BOOK  VI.-EQUATIONS  REQUIRING  IRRATIONAL  OPERATIONS. 

Ciiaiter  1.— Equations  witu  Two  Terms  only.  170.  Solution  of  a 
Binomial  Equation,  170.  Special  Forms  of  Binomial  Equations, 
171.     Positive  and  Negative  Roots,  172. 

Cuapter  II.— Quadratic  Equation.s,  174.  Solution  of  a  Completo 
Quadratic  Equation,  175.  Equations  wliicli  may  be  reduced  to 
Quadratics,  179.  Factoring  a  Quadratic  Equation,  184.  Equations 
having  Imaginary  Roots,  188. 

Chapter  III. — Reduction  of  Irrational  Equations  to  the  Noumal 
Form,  189.     Clearing  of  Surds,  189. 

Chapter  IV. — Simultaneous  Quadratic  Equations,  193. 


tios,  132. 

n,  134.    The 


BOOK   VII.— PROGRESSIONS. 

Chapter  I. — Arithmetical  Progression,  200.  Problems  in  Pro- 
gression, 202. 

Chapter  1 1. —Geometrical  Progression,  207.  Problems  of  Geo- 
metrical Progression,  208.  Limit  of  the  Sum  of  a  Progression, 
211.    Compound  Interest,  217. 


X  CONTENTS. 

PAirr    II.-ADVANCED    COURSE. 

BOOK  VIII.— RELATIONS  BETWEEN  ALGEBRAIC  QUANTITIES. 

Functions  and  their  Notation,  331.  Equations  of  the  First  Degree 
between  Two  Variables,  334.  Notation  of  Functions,  330.  Func- 
tions of  Several  Variables,  3533.  Use  of  Indices,  333,  Miscellaneous 
Functions  of  Numbers,  335. 

BOOK   IX.— THE  TIIEJRY   OF  NUMBERS. 

Chaptek  I,— The  Divisibility  of  Numbers,  338.  Division  into  Prime 
Factors,  839,  Common  Divisors  of  two  numbers,  340.  Relations 
of  numbers  to  their  Di^ts,  345.  Divisibility  of  Numbers  and  their 
Digits,  845.  Prime  Factors  of  Numbers,  3^18.  Elementary  Theorems, 
351.     Binomial  Coellicionts,  351.     Divisors  of  a  Number,  254. 

Chai'TEK  II. — Op  Continted  Fractions,  258.  Relations  of  Converging 
Fractions,  367.     Periodic  Continued  Fractions,  370. 


^i 


BOOK  X.— THE  COMBINATORY  ANALYSIS. 

Chapter  I.— P^iRMUTATiONS,  373.  Permutation  of  Sets,  275.  Circular 
Permutations,  877.  Permutations  when  Several  of  the  Things 
are  Identical,  379.  The  two  classes  of  Permutations,  381.  Sym- 
metric Functions,  884. 

CnAPTER  II, — Combinations,  285.  Combinations  with  Repetition, 
387.  Special  Cases  of  Combinations,  289.  The  Binomial  Theorem 
when  the  Power  is  a  Whole  Number,  396. 

Chapter  III,— Theory  of  Probabilities,  299.  Probabilities  depend- 
ing upon  Combinaiions,  300.  Compound  Events,  305,  Cases  of 
Unequal  Probability,  310.  Application  to  Life  Insurance,  316. 
Table  of  Mortality,  318. 


BOOK  XL— OF  SERIES  AND  THE  DOCTRINE  OF  LIMITS. 
Chapter  I.— Nature  of  a  Series,  321.    Notation  of  Sums,  324 


CONTENTS. 


XI 


UESE. 

lAIC  QUANTITIES. 

»f  tli(3  First  Decree 
mctious,  230.  Fuuc- 
,  283.    Miscellaneous 


dBERS. 

Division  into  Prime 
jors,  340.  Relations 
f  Numbers  and  their 
ementary  Theorems, 
L  Number,  254. 

itions  of  Converging 
270. 

\LYSIS. 

Sets,  275,  Circular 
eral  of  the  Things 
itations,  281.     Sym- 


i 


CUAPTEU  II.— Development  in  Powers  of  a  V^vkiabij;,  3,N», 
Method  of  Indeterminate  Coefficients,  327.  Multirillcaticn  of  Two 
Infinite  Series,  333. 

CuAPTEit  III.— Summation  of  Series.  Of  Figurate  Numl)ers,  336, 
Enumeration  of  Triangular  Piles  of  Shot.  33!>.  Sum  of  the 
Similar  Powers  of  an  Arithmetical  Progression,  311.  Otlier  Series, 
345.    Of  Difl'erences,  350.    Theorems  of  Diii'erences,  355. 

CuAPTER  IV. — The  Doctrine  of  Limits,  358.  Notation  of  the 
Method  of  Limits,  301.     Properties  of  Limits,  304. 

CuAiTER  V. — TuE  Binomial  and  Exponential  Theorems.  The 
Binomial  Theorem  for  all  values  of  the  Exponent,  30y.  The 
Exponential  Theorem,  373. 

Cuapter  VI, — Logarithms,  378.  Properties  of  Logarithms,  378.  Com- 
parison of  Two  Systems  of  Logarithms,  384. 

BOOK  XII.— IMAGINARY  QUANTITIES. 

Chapter  I. — Operations  with  the  Imaginary  Unit,  301.  Addi- 
tion of  Imaginary  Expressions,  393.  Multiplication  of  Imaginary 
Quantities,  393.  Reduction  of  FuLctious  of  i  to  the  Nonnal 
Form,  390. 

Chapter  II,— The  Geometrical  Reprei^^entation  of  Imaginar? 
Qu.vntities,  404. 


3    with    Repetition, 
I  Binomial  Theorem 


robabilities  depend- 
nts,  305.  Cases  of 
ife  Insurance,  316. 


E  OP  LIMITS. 
of  Sums,  324 


BOOK   XIII.— THE  GENERAL  THEORY  OF   EQUATIONS. 

Every  Equation  has  a  Root,  416.  Number  of  Roots  of  General 
Equation,  418.  Relations  between  Coefficients  and  Roots,  43'). 
Derived  Functions,  427.  Significance  of  the  Derived  Function,  430. 
Fom)s  of  the  Roots  of  Equation,  43 1.  D(;composition  of  Rational 
Fractions,  433.  Greatest  Common  Divisor  of  Two  Functions,  438. 
Transformation  of  Equations,  443.  Resolution  of  Numerical  Equa 
tious,  447. 


4 


FIRST   PART. 


ELEMENTARY   COURSE. 


1\ 


m 


% 


trej 

vail 
ina( 

ma 

bus 

are 

M 

tioi 
ho\ 

its 


kir 
Tin 


BOOK    I. 
THE     A  L  G E B  RAI C     E  A  N G  UA  G E . 


CHAPTER    I. 

OF    ALGEBRAIC    NUMBERS    AND    OPERATIONS. 


I 


General   Definitions. 

1.  Definition.  Mathematics  is  the  science  wMcli 
treats  of  tlie  relations  of  magnitudes. 

The  magnitudes  of  mathematics  are  time,  space,  torce, 
value,  or  other  tilings  which  can  be  thought  of  as  entirely 
made  up  of  parts. 

2.  Be/.  A  Quantity  is  a  definite  portion  of  any 
magnitude. 

Example.  Any  definite  number  of  feet,  miles,  acres, 
bushels,  years,  pounds,  or  dollars,  is  a  quantity. 

3.  Def.  Algebra  treats  of  those  relations  which 
are  true  of  quantities  of  every  kind  of  magnitude. 

4.  The  relations  treated  of  in  Algebra  are  discovered 
by  means  of  numbers. 

To  measure  a  quantity  by  number,  we  take  a  certain  por- 
tion of  the  magnitude  to  be  measured  as  a  unit,  and.  express 
how  many  of  the  units  the  quantity  contains. 

Remark.  It  is  obviously  essential  that  the  quantity  and 
its  unit  shall  be  the  same  kind  of  masrnitude. 

5.  Bef.  A  Concrete  Number  is  one  in  which  the 
kind  of  quantity  which  it  measures  is  expressed  or 
understood  ;  as  7  miles,  3  days,  or  10  pounds. 


4  77/ a;   ALUHIJUAIC    LA\(iL'Aat\ 

<>.  Def.  An  Abstract  Number  is  one  in  whicli  no 
paiiicular  kind  of  unit  is  expressed  ;  as  7,  3,  or  10. 

Kkmakk.  An  abstract  number  may  be  considered  as  a 
concrete  one  expressing  a  certain  number  of  units,  without 
res})ect  to  tlie  kind  of  units.     Tlius,  7  means  7  uniLs. 

Alj»'el)riiie   Nmnbcrs. 

7.  In  Aritliraetic,  the  numbers  begin  at  0,  and  in- 
crease witliout  limit,  as  0,  1,  2,  3,  4,  etc.  But  the 
qiuintities  we  usually  measure  by  numbers,  as  time 
and  si)ace,  do  not  really  begin  at  any  point,  but  extend 
without  end  in  opi)osite  directions. 

For  example,  time  has  no  beginning  and  no  end.  An 
epoeii  of  time  1000  years  from  Christ  may  be  eitlier  1000  years 
after  Clirist,  or  1000  years  before  Christ. 

A  heavy  body  tends  to  fall  to  the  ground.  A  body  which 
did  not  tend  to  move  at  all  when  unsupported  would  have  no 
Aveight,  or  its  weight  would  be  0.  If  it  tended  to  rise  upward, 
like  a  balloon,  it  would  have  the  opposite  of  weight. 

If  w:  have  to  measure  a  distance  from  any  point  on  a 
straiglit  line,  we  may  measure  out  in  either  direction  on  the 
line.     If  the  one  direction  is  east,  the  other  will  be  west. 

One  who  measures  his  wealth  is  poorer  by  all  that  he  owes. 
If  lie  owes  more  than  he  possesses,  he  is  worth  less  than 
nothing,  and  there  is  no  limit  to  the  amount  he  may  owe. 

8.  In  order  to  measure  such  quantities  on  a  uni- 
form system,  the  numbers  of  Algebra  are  considered  as 
increasing  from  0  in  two  opposite  directions.  Those  in 
one  direction  are  called  Positive;  those  in  the  other 

direction  Negative. 

9.  Positive  numbers  are  distinguished  by  the  sign 
+  ,  %lus ;  negative  ones  by  the  sign  ~,  minus. 

If  a  positive  number  measures  years  after  Christ,  a  negative 
one  will  mean  years  before  Christ. 

If  a  positive  number  is  used  to  measure  toward  the  right,  a 
negative  one  will  measure  toward  the  left. 


AV 


ALU EiniA W   A UMli KllS. 


f) 


11  wliicli  no 
or  10. 

tillered  as  a 
lits,  witliout 
I  nits. 


0,  and  in- 

But  the 

's,  as  time 

3ut  extend 

3  end.  An 
V  1000  years 

body  wliicli 
ild  Juivo  no 
•ise  upward, 

point  on  a 
tion  on  the 
west. 

lat  lie  owes. 
I  less  til  an 
y  owe. 

on  a  nni- 

sidered  as 

Those  in 

the  other 


^  the  sign 


a  negative 
he  right,  a 


If  a  positive  number  measures  weight,  the  negative  one 
will  imply  levity,  or  tendency  to  rise  from  the  earth. 

If  a  ])03itive  number  measures  property,  or  credit,  the  nega- 
tive one  will  Imply  debt. 

1().  'V\w  series  of  algebriiic  nnmbers  will  therefore* 
%    be  considered  as  arranged  in  the   following  way,  the 
series  going  out  to  infinity  in  both  directions. 


-S»    NEGATIVE   DIRECTION. 

Before. 
Downward, 
Debt, 
etc. 


POSITIVE   DIRECTION.     l^T 

After. 
Upward. 
Credit, 
etc. 


-tc.  -5,  -4,  -3,  -2,  -1,  0,  +1,  +2,  +3,  +4,  -t-5,  etc. 

Rem.  It  matters  not  wliich  direction  we  take  as  the 
2)ositive  one,  so  long  as  we  take  tlie  opposite  one  as 
negative. 

If  we  take  time  before  as  positive,  time  after  will  be  nega- 
tive ;  ii  we  take  west  as  the  positive  direction,  cast  will  be 
negative;  if  we  take  debt  as  positive,  credit  will  be  negative. 

11.  Positive  and  negative  numbers  may  be  conceived 
as  measuring  distances  from  a  fixed  point  on  a  straight 
line,  extending  indefinitely  in  both  directions,  the  dis- 
tances one  way  being  ])Ositive,  and  the  other  wjiy 
negative,  as  in  the  following  scheme  :  - 

etc.    —7,  -6,-5,-4,  -3,-2,-1,     0.  +1.  +2,  +3,  +4,  +5.  +6.  +7,  »^tc. 


I 


I 


I 


In  this  scale,  the  distance  between  any  two  consecu- 
tive numbers  is  considered  a  unit  or  unit  step. 

12.  Def.  The  signs  +  and  —  are  called  the  Alge- 
braic Signs,  because  they  mark  the  dirc^ction  in  which 
the  numbers  following  them  are  to  be  taken. 


*  The  student  should  copy  this  scale  of  numbers,  and  have  it  before 
'iim  in  studying  the  present  chapter. 


0 


Till':   ALUI'UillAlC   LAJSaUAGH. 


I 


The  sign  -f  may  ])C  omitted  before  positive  numbers,  wlien 
no  jmibiguity  is  tiius  j)ro(liiced.  The  numbers  2,  5,  Vi,  taken 
jilone,  signify  -\-'Z,  -+-5,  -{-I'Z.  But  the  negative  sign  must 
always  be  written  when  a  negative  number  is  intended. 

13.  Def  One  number  is  said  to  "be  Algebraically- 
Greater  tlian  another  when  on  the  preceding  scale  it 
lies  to  the  positivi^  (j'ight  lumd)  side.     Thus, 

—  2    is  algebraically  grc^ater  than     —  7 ; 

5 


(.< 


u 


u 
a 


-5. 


Alg:ebraic  Addition. 

14.  Def.  In  Algebi-a,  Addition  means  the  combi- 
nation of  quantities  according  to  their  algebraic  signs, 
the  positive  quantitic^s  being  counted  one  way  or  added, 
and  negative  ones  the  oj^posite  way  or  sul)tracted. 

15.  Def.  The  Algebraic  Sum  of  several  quantities 
is  the  surplus  of  the  positive  quantities  over  the  nega- 
tive ones,  or  of  the  negative  quantities  over  the  positive 
ones,  according  as  the  one  or  the  other  is  the  greater. 

The  sum  has  the  same  algebraic  sign  as  the  prepon- 
derating quantity. 


Example. 

The  sum  of 

+  7     and 

7 

is 

0; 

+  9       " 

I 

(( 

+  2; 

+  5       '' 

7 

a 

—  Ji. 

The  sum  of  several  positive  numbers  may  be  represented 
on  tlie  line  of  numbers,  §  11,  by  the  length  of  the  line  formed 
by  plaeing  the  lengths  represented  by  the  several  numbers 
end  to  end.  The  total  length  will  be  the  sum  of  the  partial 
lengths. 

If  any  of  the  numbers  are  negative,  the  Jilgebraic  sum  is 
represented  by  laying  their  lengths  off  in  the  opposite  direetion. 

Example  ].  The  algebraic  sum  of  the  four  numbers  9, 
—  7,  1,   —6,  "would  be  represented  thus : 


ALGEJiHAW   ADDrUON. 


I^ers,  wlien 
»  K^  takeu 

•^i^Mi  must 
k'd. 

3raically 

scale  it 


7; 

5. 


combi- 
ic  signs, 
I'  added, 
ed. 

laiitities 
le  nega- 
positive 
eater, 
prepon- 


•esented 
formed 
umbers 
partial 

snm  is 
ection. 

Jers  9, 


I 


+  0 


Ei 


-7 


— 0 


Hero,  starting  from  0,  we  measure  9  to  llio  ri<,'lit,  tlien  7 
to  tlie  left,  then  1  to  the  riglit,  then  0  to  the  left.  The  result 
Avould  bo  3  steps  to  the  left  from  0,  that  is,  —  3.  Thus,  —  3 
is  the  algebraic  sum  of  +9,  —7,  +1,  and  —  G. 

Ex.  2.  If  we  imagine  a  person  to  walk  back  and  forth 
along  the  line  of  numbers,  his  distance  from  the  starting- 
point  will  always  be  the  algebraic  sum  of  these2)arate  distances 
he  has  walked. 

Ex.  3.  A  man's  wealth  is  the  algebraic  sum  of  his  posses- 
sions and  credits,  the  debts  which  ho  owes  being  negative 
credits.  If  he  has  in  money  81000,  due  from  A  81200,  due  to 
X  $500,  due  to  Y  8350,  his  possessions  would,  in  the  language 
of  algebra,  be  summed  up  as  follows : 

Cash,  .....         +  11000 

J)\\Q  from  A,  +     1200 

Due  from  X,  ....,—      500 

Due  from  Y,  ,....—      350 

Sum  total,    .        .        .        .        +  81350 

[In  the  language  of  Algebra,  the  fact  that  he  owes  X  8500 
may  be  expressed  by  saying  that  X  owes  him  —  $500.] 

16.  Def.  To  distinguish  between  ordinary  and 
algebraic  addition,  the  former  is  called  Numerical  or 
Arithmetical  addition. 

Hence,  the  numerical  sum  of  several  numbers 
means  their  sum  as  in  aiithmetic,  without  regard  to 
their  signs. 

17.  Rem.  In  Algebra,  whenever  the  word  su7n 
is  used  without  an  adjective,  the  algebraic  sum  is 
understood. 


8  77//';   A/M/aUiAfC    LANdUAdK 

A I  jifobrn  iv    Subtract  ioii. 

1 8.  Memorandum  of  arithmi'tical  dejinitionfi  and  operations. 
Tlic  Subtrahend  is  the  (niiintity  to  l)o  siibtriictod. 

Tlie  Minuend  is  the  qiumtity  from  which  the  subtriihciid 
is  tukt'ii. 

'V\\v  Remainder  or  Difference  is  Avlitit  is  left. 

If  wo  subtract  4  from  7,  tlio  .•enmiiKU'r  3  is  the  numbor  of 

unit  steps  on  the  scale  of  iiim.bers  (§11)  from  4-4  to  +7. 

This  is   true  of  any  arithmetical  dilference  of  numbers.     In 

Algel)ra,  the  o})eration  is  generalized  as  follows: 

19.  Dcf.  The  Algebraic  Difference  of  two  nnm- 
bors  is  reprosented  by  the  distance  from  one  to  the 
other  on  the  scale  of  numbers. 

The  number  from  which  we  measure  is  the  Subtra- 
hend. 

Tliat  to  which  we  measure  is  the  Minuend. 

If  the  minuend  is  algebraically  the  greater  (§  13), 
the  difference  is  positive. 

If  the  minuend  is  less  than  the  subtrahend,  tlie  dif- 
ference is  negative. 

In  Arithmetic  we  cannot  subtract  a  greater  number  from  a 
less  one.  But  there  is  no  such  restriction  in  Algebra,  because 
algebraic  subtraction  does  not  mean  taking  away,  but  finding 
a  difference.  However  the  minuend  and  subtrahend  may  be 
situated  on  the  scale,  a  certain  number  of  spaces  toward  the 
right  or  toward  tlie  left  will  ahvays  carry  us  from  the  subtra- 
liend  to  the  minuend,  and  these  spaces  make  up  the  difference 
of  the  two  numbers. 

30.  The  general  rule  for  algebraic  subtraction  may  be 
deduced  as  follows :  It  is  evident  that  if  we  pass  from  the 
subtrahend  to  0  on  the  scale,  and  then  from  0  to  the  minuend, 
the  algebraic  sum  of  these  two  motions  will  be  the  entire  space 
between  the  subtrahend  and  minuend,  and  will  therefore  be 
the  remainder  required.  But  the  first  motion  will  be  equal  to 
the  subtrahend,  but  positive  if  that  quantity  is  negative,  and 
vice  versa,  and  the  second  motion  will  be  equal  to  the  minuend. 


1 


He 

f-igi 
mil 

Sn 


% 


(» 


ALUEIiRA  KJ   MUL TIl'L/CA  TION. 


0 


I  operations. 

cd. 

subtniliond 


nnm])or  of 
f  4  to  +7. 
nibtTS.     In 


.wo  nnm- 
le  to  the 

^  Subtra- 


er  r§  13), 
,  tlie  clif- 

)ci'  from  a 
fi,  bocanse 
lit  findir.g 
d  may  be 
•ward  the 
10  siibtra- 
diffcreiice 

may  l)e 

from  tlie 

Qiiiiuend, 

ire  space 

refore  be 

equal  to 
tive,  and 
ninuend. 


Hence  tb.c  remainder  will  be  found  by  cbanfrin.i;  the  algebraic 
fiign  of  the  Mubtraliend,  and  tlu-n  adding  it  algebraically  to  tho 
minuend. 

EXAMPLES. 


Subtracting   +5  from   -f-  S,  the  dilTerence  is      S  —  5  —  :?. 

<'__5_S  =  —  i:;. 
''      5H   8=  -I-  1:5. 
''  0,    '' 

<<  (\      K  ii  a 


(t 

(< 
n 
ft 


+  8 
—  8 

-  13 


5. 


a 
n 


—  v.). 
0,  ''        ''        ''  +1;}. 

21.  By  comparing  algebraic  addition  and  subtraction,  it 
Avill  l)e  seen  that  to  siibtract  a  ])ositive  numl)er  is  the  same 
thing  as  to  add  its  T'egative,  and  rice  versa,     Thus, 

To  subtract  5  from  8  gives  tlie  same  result  as  to  add  —  5 
to  8,  namely  3. 

To  su])tract  —  5  from  8  gives  8  -f-  5,  namely  13. 

Henc(\  algebraic  subtraction  is  oquivahmt  to  the 
algebraic  addition  of  a  number  with  the  oi)posite 
algebraic  sign.  Algebraists,  therefore,  do  not  consider 
subtraction  as  an  operation  distinct  from  addition. 

Algebraic   ^lultiplication. 

22.  Memorandum  of  arithmetical  definitions. 

The  Multiplicand  is  the  quantity  to  be  multiplied. 
The  Multiplier  is  the  number  by  which  it  is  multiplied. 
The  result  is  called  the  Product. 

Factors  of  a  number  arc  the  multiplicand  and  multiplier 
which  produce  it. 

23.  To  multiply  any  algebraic  quantity  by  a  posi- 
tive whole  number  means,  as  in  Arithmetic,  to  take  it  a 
number  of  times  equal  to  the  multiplier. 

Thus,  4x3zrr       4  +  4  +  4=+ 12; 

-4x3  =  -4  —  4  —  4=  —  12. 

The  product  of  a  negative  multiplicand  by  a  positive 
multiplier  will  therefore  be  negative. 


10 


THE  ALdHnitMC   fsANdUAHR. 


)i\.  It'  th<'  iimltiplior  is  no^ativo,  the  sign  of  tlic 
yn'odiict  will  bu  the  ()p))OHite  of  wliat  it  would  be  if  tin* 
luultiplicr  were  positive. 

TiiiH,  4.4  X  -3  =   -  12; 

—4  X   -.'3  =    +  VZ, 

The  prodtict  of  two  negative  fac5t()rs  is  tben^fore 
positive. 

^5,  The  Jiiost  f5imi)U'  Wiiy  of  mastoriiig  tiio  use  of  algi'bniic 
fiigns  in  niitltipliciition  is  to  think  of  thu  nign  —  ns  nicmiiii.ijf 
opposite  in  directioii.  '^riiiis,  in  ^  11,  — 4  is  opposite  in 
direction  to  +  4,  the  direction  being  tiiut  from  0.  If  we  mul- 
tiply this  negative  factor  hy  a  !iegative  multiplier,  the  direction 
will  he  the  opposite  of  negative,  that  is,  it  will  he  posi/ii'c.  A 
third  negative  factor  will  make  the  product  negative  again,  a 
fourth  one  i)ositivc,  and  so  on.     For  example, 

-3  X  -4  =  +  12; 

_2  X  -3  X  -4  =  -2  X  +12  =  -24; 

—3  X  —2  X  -3  X  -4  =  -3  X  -24  =  +  72 ; 

etc.  etc. 

Ilencc, 

36.  Tlieorem.  The  continued  product  of  an  even 
number  of  negative  factors  is  positive  ;  of  an  odd  num- 
"ber,  negative. 

Rem.  Multiplying  a  number  by  —1  simply  changes 
its  sign. 


Thus, 


4-4  X  —1  =  —4; 
-4  X  -1  :=  4-  4. 


EXERCISES. 

Find  the  algeljraic  sums  of  the  following  quantities  : 

1.  4  _  G  +  12  —  1  —  18. 

2.  —0—3  —  8. 

3.  _  G  —  10  —  9  +  34. 

4.  Subtract  the  sum  in  Ex.  3  from  the  sum  in  Ex.  2. 

5.  Subtract  the  sum  5  —  6-f3  —  1  —  16,  from  the  sum 
2  —  7—4-1-8. 


A L n En liA W   DIVISION. 


n 


^\^n   of  {\\o. 
id  bcif  the 


S   tluTelbre 


i  of  al<^c'l)riii(; 
-  as  nu'imiiiL^ 

<)])j)osilo   ill 

If  wo  nitil- 

tlio  direction 

positive.  A 
itivu  a«3'iiiii,  a 


1    '*"> 


of  an  even 
1  odd  num- 

)ly  clianges 


itics : 


Ex.  2. 

»m  the  sum 


6.  Siil)lmct  the  sum  5  —  04-3  —  1  —  10,  from  tlii'  sum 
;  _  ;j  _  H  +  4. 

7.  Form  tljo  product   —  7  x  H. 

8.  Form  the  product   —  H  x  7. 

9.  Form  tilt'  product  (i  x  —  5  x  7  x  —  4. 

10.  Form  the  product   —  f)  x  —  1 1  X  H  x  —  2. 

11.  Form  the  product   — Ix—  Ix-lx—  1. 

12.  Subtract  the  sum  in  Jv\.  1  from  tlie  sum  in  Kx.  :),  and 
multiply  the  remainder  hy  the  sum  in  Kx.  2. 

I  ^  Subtract  H  from  ~  '.),  —  3  from  —1,-1  from  H,  and 
find  the  sum  of  the  three  remain(h>rs. 

14.  Subtract  7  from  —  U  and  tho  remainder  from  'Z,  and 
multii)ly  tho  result  by  tho  j)roduct  in  Kx.  7. 


Alpf(^brai<^    Division. 

27.  McNionitnlmii  of  arilhnu'tical  drfntiHons. 

The  Dividend  is  the  (piantity  to  be  divided. 
The  Divisor  is  the  number  by  wliicii  it  is  divided. 
The  Quotient  is  the  result. 

*^8.  Jlftle  of  Signs  in  Dlrl.slon.  The  reqnirement 
of  division  in  Algebra  is  the  same  as  in  Aiitlinietic  ; 
nanndy, 

T/ie  ])}'o(hi,ct  of  the  (pwticnt  hy  the  divisor  Diusb  he 
Cf/icnl  to  the  dividend. 

In  Algebra,  two  (piantities  are  not  cfpud  unless  they  have 
the  same  algebraic  sign.     Therefore  the  i)roduct, 

quotient  x  divisor 

must  have  the  same  algebraic  sign  as  the  dividend.     From 
this  we  can  deduce  the  rule  of  signs  in  division. 

Let  us  divide  G  by  2,  giving  G  and  2  both  algebraic  signs, 
and  find  the  signs  of  the  quotient  3 : 

+  3  X  +2  =  4-G;  therefore,  -f-G  divided  by  +2  gives  +3. 

+  3x— 2  =  — G;          "  — G       ''        *'   —2     ''      +3. 

— 3  X  +2  =:  — G;          "  — G       ''        "    +2     ''      —3. 

—  3  X  — 2  =  -fG;          ''  4-G       "        ''   —2     "'      —3. 


'!     ,t 


12 


THI^J   ALUEBliAlC   LANGUAGE. 


nonce,  the  rule  of  signs  is  the  same  in  division  as  in  mul- 
tii)lication,  namely  : 

Like  si^ns  in  dividend  and  divisor  give  +.     Unlike 


signs  give  — . 


EXERCISES, 


Execnte  the  folloAving  algebraic  divisions,  expressing  each 
result  as  a  whole  number  or  \  ulgar  fraction  • 

Dividend,  —  7  +  10  —  11  +  25  ;  divisor,  20  —  3. 
Dividend,  12  —  3  +  15  —  10  ;  divisor,  3  —  10. 
Dividend,  25  —  3G  +  G  —  20  ; 
Dividend,  —  7  x  —  8  ; 
Dividend,  5G  +  8  x  —  3  ; 
Dividend,  —  24  x  —  1 ; 


I. 

2. 

3- 

4- 

5- 
6. 

7- 


divisor,  —3  +  8. 

divisor,  —  8  +  4. 

divisor,  —  4  —  4. 

divisor,  —  3  x  —  3. 


Dividend,  —13  x  —10  x  —  8;  divisor, 


8.  Dividend,  —  1  x  —  1  ; 


divisor, 


•4x5  X  — G. 
3  X  -3. 


eh. 


nil' 


-♦■♦■♦- 


CHAPTER    II. 


ALGEBRAIC     SYM  BOLS. 


Symbols  of  Quantity. 

29.  Algebraic  quantities  may  be  represented  by 
letters  of  tlie  alx)liabet,  or  other  characters. 

The  characters  of  Algebra  are  called  Symbols. 

30.  Def.  'rhe  Value  of  an  algebraic  symbol  is  the 
quantity  which  it  represents  or  to  which  it  is  equal. 

The  value  of  a  symbol  may  be  any  algebraic  quan- 
tity whatever,  positive  or  negative,  which  we  choose  to 
assign  to  tlie  symbol. 

31 .  The  language  of  Algebra  differs  in  one  respect  from 
ordinary  language.     In  the  latter,  each  special  word  or  sign 


I 


tiiaJVti   OF  OPEUATWN. 


vs 


m  as  ill  iiml- 
4- .     Unlike 


pressing  caeli 


0-3. 
-10. 
-3  +  8. 
-8  +  4. 
-4  —  4. 

-  3  X  -  3. 
-4x5  X  — G. 

-  3  X  -  3. 


'sented  by 

ibols. 

nbol  is  the 
equal. 
)raic  quan- 
-i  choose  to 

respect  from 
•ord  or  sign 


has  a  definite  and  invariable  meaning,  wliicli  every  one  who 
uses  the  hinguage  must  learn  onee  for  all.  But  in  Algebra  a 
.^vnibol  may  stand  for  any  quantity  which  the  writer  or  spejiker 
chooses,  and  his  results  must  be  interpreted  according  to  this 


meaning. 


\\2,  The  same  character  may  be  used  to  represent  several 
(|uantities  by  applying  accents  or  attaching  numi)ers  to  it  to 
disringuisii  the  different  ([uantities.  Thus,  the  four  symbols, 
a,  a,  a",  a'",  may  represent  four  different  quiuititics.  The 
symbols  ^i,  (ti,  «.,,  ^^4,  «5,  etc.,  may  be  used  to  designate  any 
number  of  ([uantities  which  are  distinguished  by  the  small 
number  written  after  the  letter  a. 

Signs  of  Operiitioii. 

8,3.  In  Algebra,  the  signs  +,  — ,  and  x  are  used, 
as  in  Arithmetic,  to  represent  addition,  subtraction,  and 
multiplication,  these  operations  being  algebraic,  not 
numerical. 

84.  Signs  of  Addition  and  Siihtractlon.  The  com- 
bination a-\-h  means  the  algebraic  sum  of  the  quantities 
a  and  ^,  and  a  —  h  means  their  algebraic  difference. 

EXAMPLES. 


If  ^?  =  +  4  and  h  =  +  ^,  then 
If  c/  ~  +  5  and  b  =:  —  7,  then 
li  a—  —  Q>  and  h  =  + '^,    then 


a-\-J)=  +;,  a  —  h  —  +1. 
a-{-h=  —2,  a  —  h  =  +12. 
a-\-b=  —3,  a—h—  —9. 


If  rt  =:  —  (5  and  J  =  —  3,     then     rt  +  J  =  —9,  a—h  =  —3. 

The  signs  of  addition  and  subtraction  are  the  same  as  those 
used  to  indicate  positive  and  negative  quantities,  but  the  two 
a[>})lieations  may  be  made  without  confusion,  because  the 
op})osite  positive  and  negative  directions  correspond  to  the 
opposite  operations  of  adding  and  subtracting. 

,85.  Sign  of  Multiplication.  The  sign  of  multipli- 
cation, X ,  is  generally  omitted  in  Algebra,  and  when 
different  symbols  are  to  be  multiplied,  the  multiplier  is 


''    't 


14 


TUE   LANGUAGE  OF  ALGEBRA. 


written  before  the  multiplicand  without  any  sign  be- 
tween them. 

Tims,  4a     means    a  x  4. 

ax        "         X  X  a. 
Mthmy        "         1/  X  }n  X  h  X  a  X  3. 

If  numbers  are  used  instead  of  symbols,  some  sign  of  mnl- 
tiplication  must  be  inserted  between  them  to  avoid  confusion. 
Thus,  34  would  be  confounded  witli  the  number  thirty-four. 
A  simple  dot  is  therefore  inserted  instead  of  the  sign  x . 

Thus,  3.4  =:  4x3  =  13. 

3- '2.2  =  72. 
1.2-3.4.5  =  120. 
I.2.3.4-5.G  =  720. 
The  only  reason  why  the  point  is  used  instead  of   x,  is 
that  it  is  more  easily  written  and  takes  up  less  space. 

36.  Division  in  Algebra  is  sometimes  represented 
by  the  symbol  -^,  the  dividend  being  placed  to  the  left 
and  the  divisor  to  the  right  of  this  symbol. 

Ex.     a  -^h  means  the  quotient  of  a  divided  by  h. 

But  division  is  more  generally  represented  by  writing 
the  dividend  as  the  numerator  and  the  divisor  as  the 
der  ominator  of  a  fraction. 

Ex.     The  quotieiit  of  a  divided  by  h  is  w^'itten     • 

It  is  shown  in  Arithmetic  that  a  fraction  is  equal  to  the 
quotient  of  its  numerator  divided  by  its  denominator  ;  hence 
this  expression  for  a  quotient  is  a  vulgar  fraction. 

37.  Powers  and  Exponents.  A  Power  :>f  a  quan- 
tity is  the  product  obtained  by  taking  that  quantity  a 
ceru..:n  number  of  times  as  a  factor. 

Def.  The  Degree  of  the  power  means  the  number 
of  times  the  quantity  is  taken  as  a  factor. 

If  a  q  aantity  is  to  be  raised  to  a  power,  the  result 
may,  in  accordance  with  the  rule  for  multiplication,  be 


.,?!■ 


SIGNS   OF  OPERATION. 


16 


ly  sign  Ibe- 


i  X  3. 

sign  of  mul- 
id  confusion. 
I'  tliirty-fuur. 
ign  X. 


}ad  of   X,  is 
ace. 

represented 
i  to  the  left 


I  by  writing 
dsor  as  the 

a 

'*• 

equal  to  the 
lator  ;  hence 

c>f  a  qnan- 
quantity  a 

the  number 

,  the  result 
lication,  be 


or 


aaaad  ; 


exi)ressed  by  writing  the  quantity  the  required  number 

of  times. 
m      Examples.     The  fifth  power  of  a  may  be  written 
If  axa  xaxaxa 

and  the  fourth  power  of  7, 

To  save  repetition,  the  symbol  of  which  the  power  is 
to  be  expressed  is  written  but  once,  and  tlie  number  of 
times  it  is  taken  as  a  factor  is  written  in  small  ligures 
after  and  above  it. 


7.7.7.7  =  2-101. 


Thu: 


aaaaa    is  written 

7.7.7.7     "        " 

XXX       "  " 


a"" 


7^ 


D(f.  A  figure  written  to  indicate  a  power  is  called 
an  Exponent. 

Def.  The  operation  of  forming  a  power  is  called 
Involution. 

.*58.  Hoots.  A  Root  is  one  of  the  equal  factors 
into  which  a  number  can  be  divided. 

D(f.  The  iiguie  or  letter  showing  the  number  of 
equal  factors  into  which  a  quantity  is  to  be  divided  is 
called  the  Index  of  the  root. 

The  square  root  of  a  symbol  is  expressed  by  writing 
the  sign  ^  (called  root)  before  it. 

Ex.  I.     V'l'J  means  the  square  root  of  49,  that  is,  7. 
Ex.  2.     "s/x    means  the  square  root  of  x. 

Any  other  root  than  the  square  is  represented  by 
writing  its  index  before  the  sign  of  the  root. 

Ex.  I.     v'x  means  tlie  rube  root  of  .r. 
Ex.  2.     \/ X  means  the  fourth  root  of  x. 

Def.  The  operation  of  extracting  a  root  is  called 
Evolution. 

31).  The  operations  of  Addition,  Subtraction,  Multi- 
])lication,  Division,  Involution,  and  Evolution,  are  the 
six  fundamental  operations  of  Algebra. 


10 


777^^   ALGEBRAIC  LANGUAGE. 


4().  Def.  An  Algebraic  Expression  is  any  combi- 
nation of  algebraic  symbols  made  in  accordance  witli 
the  foregoing  principles. 


EXERCISES. 


In  the  following  expressions,  suppose  «  =  —  7,  ^  =  —  5, 
c  =  0,  VI  =  3,  n  =  4,  J)  ■=:  9,  and  compute  tlieir  nnmerictil 
values. 


I. 

r/  +  Z*  +  ?/?.  +  p. 

2. 

a  +  ^y?.  +  ;?. 

1)1  —  n  —  a  —  b. 

4- 

71  -f  /;  —  yyi  —  a. 

5- 

'.](t  —  ))i  -}-  b  —  2n. 

6. 

'2a       Ty;  +  2b       m. 

7- 

''ionip, 

8. 

mncp. 

9- 

bmn. 

10. 

blip. 

II. 

ab))ip. 

12. 

2'^abHp. 

IS- 

a)n ^-  bn. 

14. 

am  —  bn. 

IS- 

bp  —  an. 

16. 

Gp  +  an. 

17- 

)i^p  +  m%. 

18. 

vi^n  —  ap^. 

19. 

a^  +  b'\ 

20. 

.73  +  b^ 

2  1. 

(r       b\ 

22. 

a^ni  —  b^n. 

23- 

a'^b'^  —  m^n-. 

24. 

(1%^  —  b'^nA 

25- 

ab^  +  aVj. 

26. 

ab^  —  a^. 

'^  M 

ab  +  mn 

^Q 

ac  —  bp 

27 


29. 


ab  —  mn 
2inhi^  —  10^3 

. —  « 

})  —  bcm 


30. 


bn  —  7np 
ab  —  7np 


m 


n 


In  the  following  expressions,  suppose  a  =  S,  b  =:  —  3,  and 
X  to  have  in  succession  the  fifteen  values  —7,  — 6,  — 5,  etc., 
to  I-  7,  and  compute  the  fifteen  corresponding  values  of  each 
expression : 

a  -\-  bx 


31.     x"^  -{-  bx  -\-  a. 


32. 


a  —  bx 

Arrange  the  results  in  a  table,  thus  : 

Expression  31  =  78  ; 

"  "  =  62  ; 

"  "  =  48. 

etc.  etc. 


X  =  -i  ; 

X  r=  —  0  ; 

ic  =  — 5  ; 

etc. 


Exp.  32  =  -  fl- 
etc. 


-1 

•■Pi 


the 


COMPOUND   EXPRESSIONS. 


17 


iiy  combi- 
iance  witli 


,  ^  =  —  5, 
numerical 


-  a. 
?  —  m. 


:  —  3,  and 
.  —5,  etc., 
es  of  each 


=  -ff. 


:g. 


CHAPTER     III. 


# 


m  FORMATION    OF    COMPOUND    EXPRESSIONS. 

Si 


g  Fundaiiieiital    I'rinciples. 

^  41.  The  following  are  two  fundamental  principles  of 
tlie  algebraic  language : 

*  First  Frmciple.  Every  algebraic  expression,  how- 
ever coni])lex,  represents  a  quantity,  and  may  be 
operated  upon  as  if  it  were  a  single  symbol  of  that 
quantity. 

Second  Prhiciple.     A  single  symbol  may  be  used 
to  represent  any  algebraic  expression  whatever. 

I       4'-^.  When  an  expression  is  to  be  operated  upon  as 
'  a  single  quantity,  it  is  enclosed  between  parentheses, 

but  the  parentheses  may  be  omitted,  when  no  ambiguity 

or  error  will  result  from  the  omission. 

I  Example.  Let  us  have  to  subtract  /;  from  a,  and  multiply 
the  remainder  by  the  factor  m.  The  remainder  will  be  ex- 
pressed by  a  —  h,  and  if  we  write  the  product  of  this  quantity 
by  m,  in  the  way  of  §  35,  the  result  will  be 

ma  —  h. 
But  this  will  mean  h  subtracted  from  ma,  which  is  not  what 
we  want,  because  it  is  not  a,  but  a  —  b  which  is  to  be  multi- 
plied by  m.     To  express  the  required  operations,  we  enclose 
a  —  b  in  brackets  or  parentheses,  and  write  m  outside,  thus  : 

m  [a  —  b). 

NUMERICAL      EXAMPLES. 

T(8-:i)  =r  r-O  r^  42;     but     7-8  -  2  ==:  5G  —  2  =  54. 
12(3  +  4)  =  12-7  =  84. 
(G  +3)  (2  4-  n)  =  9-8  =  72. 
(7  _  4)  (1-5)  (2+  7)  =  3  X  -4-9  =  -108. 
2 


18 


TUE  LANGUAGE   OF  ALGEBRA. 


Example  2.  Suppose  that  tlie  expression  a  —  h  -\-  c  is  to 
be  added  to  /;/,  subtracted  from  m,  multiplied  l)y  ;//,  divided 
by  III,  raised  to  the  third  power,  or  have  the  cube  root  extracted. 
The  results  will  be  written: 


Added  to  m. 
Subtracted  from  iiiy 
Multiplied  by  m, 

Divided  by  ?«, 

Cubed, 

Cube  root  extracted, 


VI  -\-  {a  —  h  -\-  c). 
m  —  {a  —  b  -\-  c). 
m  {a  —  b  +  c). 
{a  —  b  -\-  c) 

{a  —  b  +  c)^ 
\/{a  —  b  +  c). 


There  are  two  of  these  six  cases  in  Avhieh  the  parentheses 
arc  unnecessary,  although  they  do  no  harm,  namely,  addition 
and  division,  because  in  the  case  of  addition, 

m  -[-  {a  —  b  -\-  c) 
is  the  same  as  m  -\-  a  —b  ■\-  c. 

[For  example,     10  +  (8  -  5  +  4)  =  10  +  7  ==  17, 
and  10+8—5  +  1     =17  also.] 

Again,  in  the  case  of  the  fraction,  it  will  be  seen  that  it  has 
exactly  the  jame  meaning  with  or  without  the  parentlieses. 

43.  An  algebraic  expression  having  parentheses  as 
a  pai-t  of  it  may  be  itself  enclosed  in  parentheses  with 
other  expressions,  and  this  may  be  repeated  to  any 
extent.  Eiich  order  of  parentheses  mnst  then  be  made 
larger  or  thicker,  or  different  in  shape  to  distinguish  it. 

Examples,  i.  Suppose  that  we  have  to  subtract  a  from 
I,  the  remainder  from  c,  that  remainder  from  d,  and  so  on. 
We  shall  have, 

b  —  a. 
c —  {b  —-  a). 
d  _  [6-  _  {b  -  d)\ 
c-Xd-lc-ib-fi^W. 
f^\_e-\cl-ic-{b-a)-\\\ 


First  remainder. 

Second, 

Tiiird, 

Fourth, 

Fifth. 


'•', 


1 


DEFINITIONS. 


19 


-  b  -\-  c  \s  io 
y  m,  divided 
ot  extracted. 


3  paron theses 
ely,  addition 


=  17, 
I 

n  tliat  it  has 
eiitheses. 

^theses  as 
theses  with 
ted  to  any 
n  be  made 
tingiiish  it. 

tract  a  from 
,  and  so  on. 


f^ 


2.  Suppose  that  we  have  to  multiply  the  dilTerenco  of  the 
quantities  a  and  b  hy  ;;  and  subtract  the  product  i'roni  ni.  The 
result  or  reniaindei  will  be 

7n  —p(a—  b). 

Suppose  now  that  we  have  to  multiply  this  result  by  p-\-q. 
We  must  enclose  both  factors  in  parentheses,  and  the  result 
will  then  be  written  : 

(i>  +  q)[m-2^{a-b)'\. 

EXERCISES. 

In  the  following  expressions,  suppose 
r/  —  —  1,     Z*  =  3,    m  =  5.     X  =z  —  3,  —  1,  -f  1,  4-  3, 
and  calculate  the  foUx'  values  of  each  expression  which  result 
from  giving  x  the  above  four  values  in  succession. 

X  {x  —  a)  {x  —  2a)  {x  —  3a) 


I. 


2.       - 


3- 

4- 


l.^-3--4 

[a  {b  —  x)  —  b{a  —  x)f 

m  [b  —  x)  -\-  b  {m  —  ;/•) 

[ax  +  b  {x  —  rt)2  4-  71)  {x  ■ 
l^{mx^  +  b)  -  ^{mx^ 


X  —  m 


'  -^  X  -\-  m 


]. 


XoTE.     VVlien  the  square  root  is  not  an  integer,  it  will  be  sufficient 
to  express  it  without  computing  it  in  full. 
Til  us,  for  a*  =  —  8,  we  shall  have 

^(wa;2  +  5)  _  ^{rriT^i  -  6)  =  y'^S  -  V^^- 
This  is  a  sufficient  answer  without  extracting  the  roots. 

Definitions. 

44.  Coefficient.  Any  number  which  multiplies  a 
quantity  is  called  a  Coefficient  of  that  quantity.  A 
coefficient  is  therefore  a  multiplier. 

Example.     In  the  expression  4r;^.r, 

4  is  the  coefficient  of  ahx, 
^a    "  "  "   bx, 

Aab     "  "  "   X. 


^^-^ 


20 


THE  LANGUAGE  OF  ALGEBRA. 


Dr/.    A  Numerical  Coefficient  is  a  simple  number, 
as  4,  in  the  above  exaiii])le. 

I)('/.     A  Literal  Coefficient  is  one  containing  one 
or  more  letter's  used  as  algebraic  symbols. 

Rp:m.     Any  quantity  may  be  considered  as  having 
the  coefficient  1,  bec^ause  Ix  is  the  same  as  x. 

Reciprocal.    The  Reciprocal  of  a  number  is  unity 
divided  by  that  number.     In  the  language  of  Algebra, 

1 


lleclproral  of  X 


N 


Formula.  A  Formula  is  an  exi)ression  used  to 
show  how  a  quantity  is  to  be  expressed  or  calculated. 

Term.  When  an  expression  is  made  up  of  several 
parts  connected  by  the  signs  +  or  — ,  each  of  these 
])arts  is  called  a  Term. 

Example. — In  tlie  expression, 

a  +  bx  4-  '^mx^, 
lliere  are  three  terms,  a,  bx,  and  3mx^. 

When  several  terms  are  enclosed  between  parentheses,  so 
as  to  be  operated  on  as  a  single  symbol,  they  form  a  single 
term. 

Thus,  the  expression 

{a  +  bx  4-  3mx^)  (a  +  b) 

(^^Tl/)!^  —Y) 

forms  but  a  single  term,  though  both  numerator  and  denom- 
inator are  each  a  product  of  several  terras.  Such  expressions 
may  be  called  compound  terms. 

Aggregate.  A  sum  of  several  terms  enclosed  be- 
tween parentheses  in  order  to  be  operated  upon  as  a 
single  quantity  is  called  an  Aggregate. 

Algebraic  expressions  are  divided  into  monomials 
and  polynomials. 

A  Monomial  consists  of  a  single  term. 


DEFINITIONS. 


21 


e  number, 


lining  ono 


as  liaving 


pr  is  unity 
'  Algebra, 


1  used  to 
Iculated. 

of  several 
L  of  these 


'n theses,  so 
m  a  single 


md  denom- 
3xpressions 

losed  be- 
pon  as  a 

onomials 


A  Polynomial  consists  of  more  than  one  term. 
A  Binomial  is  i\  polynomial  of  two  t<'rms. 
A  Trinomial  is  a  polynomial  of  three  tenns. 

Note.  The  last  three  words  are  coinnioiily  a|)})li(.'d  only 
to  suras  of  sinii)ie  terms,  formed  of  single  symbols  or  i)roducts 
of  single  symbols. 

Entire.  An  Entire  Quantity  is  one  which  is  ex- 
pressed without  any  denominator  or  divisor,  as  2,  3,  4, 
etc.  ;  a,  5,  .r,  etc.  ;  2(ih,  2mp^  ah  {x  —  ?/),  etc. 

A  Theorem  is  the  statement  of  any  general  truth. 

45.  Other  Alyehraic  Signs.  Besides  the  signs  al- 
ready defined,  others  are  of  occasional  use  in  Algebra. 

>,  the  Sign  of  Inequality,  shows  when  placed  be- 
tween two  quantities,  that  the  one  at  the  open  end  of 
the  angle  is  the  greater. 

'^         Ex.  I.     rt  >  J  means  a  is  greater  tlian  h. 

Ex.  2.  m  <Cx  <Cn  means  x  is  greater  than  m,  but  less 
than  71. 

: ,  another  Sign  of  Division,  is  placed  between  two 
quantities  to  express  their  ratio. 

*         Thus,  a  :  b  means  the  ratio  of  a  to  b,  or  the  quotient  of  a 
divided  by  b. 

i 

.'.  means  Hence,  or  Consequently;  as, 

rt  +  2  =  5  ;  .-.     a  -~  3. 

QC  means  a  quantity  infinitely  great,  or  Infinity. 

,  tlie  Vinculum,  is  sometimes  placed  over  an 
aggregate  to  include  it  in  one  mass,  in  lieu  of  paren- 
theses. 


Ex.     a  —  b  c  —  d  h  the  same  as  {a  —  b)  {c  —  d). 

It  is  mostly  used  with  the  radical  sign.    We  often  write 

\^a  -{-  b  -\-  c    insteafl  of     V{n  -]-  b  -\-  c). 


22 


TUE  LANGUAGE   OF  ALGEBRA. 


CHAPTER     IV. 

CONSTRUCTION    OF    ALGEBRAIC    EXPRESSIONS. 

4(>.  All  operations  upon  ulgobruic  (piuiititic's,  liowovor 
roinplux,  consist  in  eombiiuitiuns  of  tlic  cleiiicnhiry  operations 
ulreiuly  descrilK'd.  The  rcaiilt  of  each  single  operation  will  he 
jin  iig^M'e^^tite,  ;i  product,  ii  quotient,  or  ji  root,  and  every  such 
result  may,  in  subsequent  opci-ations,  be  o])erated  upon  as  a 
single  sytnbol.  There  are  only  three  '"'ses  in  which  an  expres- 
yion  needs  any  modification  in  order  to  be  operated  upon, 
namely : 

Case  I.  An  aggregate  must  be  enclosed  in  parentheses,  if 
any  other  operations  than  addition  or  division  are  to  be  jier- 
fornied  upon  it.     (§  42.) 

Case  II.  When  a  product  is  to  be  raised  to  a  power,  or  to 
liave  a  root  extracted,  it  may  be  enclosed  in  parentheses  in 
order  to  show  that  the  operation  extends  to  all  the  factors. 

If  wc  take  the  product  abc,  and  write  an  exponent,  3  for 
instance,  after  it  thus,  ahc^y  it  would  ap[)ly  only  to  c,  and 
Avoulv'.  mean  a  x  h  x  c^.  So  with  the  radical  sign  ;  ^abc 
might  mean  only  ^((  xbxc.  To  indicate  that  the  power 
or  root  is  that  of  the  product  as  a  whole,  we  may  enclose  it 
in  parentheses,  thus  : 

Square  root  of  abc  =  ^^{ahc). 
Square  of  abc  =  (abc)'^. 

But  a  root  sign  is  commonly  made  to  include  the  whole 
product  by  simply  extending  a  vinculum  over  all  the  factors 
of  the  product,  thus :    Square  root  of  abc  =  Vctbc. 

Case  III.  If  negative  quantities  are  to  be  multiplied, 
merely  writing  them  after  each  other  would  lead  to  mistakes. 
Thus,  the  product  ax — bx — c,  if  written  without  the  x 
sign,  Avould  be  a  —  b  —  c,  and  would  not  mean  a  product  at 
all.     But,  l)y  enclosing  —b  and  —c  in  parentheses,  we  have 

a{-b){-c), 
which  would  correctly  express  the  product  required. 


br 


COXSTIiUcriON  OF  ALinmUAlC  KXl'RESHIOm.      28 


SSIONS. 

k's,  however 
ry  0})cruti()ns 
tition  will  be 
I  every  such 
<l  upon  as  a 
•h  an  expres- 
jrated   upon, 

irentlieses,  if 
L'e  Lo  be  pcr- 

power,  or  to 
.re n theses  in 
e  factors. 
Dnent,  3  for 
y  to  c,  and 
sign  ;  ^ahc 
t  the  power 
Ely  enclose  it 


le  the  whole 
the  factors 

> 

multiplied, 
to  mistakes. 
lOut   the    X 

product  at 
3,  we  have 

d. 


17.  The  following  example  will  siiow  how  ()[)eration8  may 
be  combined  to  any  extent. 

The  ((iiantity  f/  is  to  be  subtracted  from /y,  and  the  dilTcr- 
cnce  multiplied  by  y,  forming  a  product  P.^  The  ([uotient  of 
■p  _  /•  divided  by  7  is  to  Ije  mullii)lied  by  ///,  aiul  the  i)r(»duet 
subtracted  I'roni  l\  The  dillerence  is  to  form  the  numerator 
.\' oj" a  fraction.  To  form  the  denomiiuitor,  /,*  is  to  be  added 
to  ((  and  subtracted  from  it,  and  the  i)roduct  Qo['{\\q  sum  and 
dill'ereuce  Formed.  The  quantity  q  is  to  be  added  to  and  sub- 
tracted from  p,  and  the  product  R  of  the  sum  and  difference 
formed.  The  (juotient  of  Q  divitled  by  R  is  to  form  the  de- 
iU)unuator  of  the  fraction  of  which  the  numerator  is  1\ 

The  f|uantity  h  subtracted  from  (i  leaves   h  —  a. 

]\Iultii»lyiug  it  by  y,  the  product  J*  is        y  {h  —  a). 


Quotient  of  p  —  r  divided  by  y 
Multiplying  it  by  m, 


p  —  r 


VI 


(1 

P  —  r , 

Q 

[If  instead  of  multiplying  the  fraction  as  a  whole  by  ni, 

we    had    mulli})lied   its  numerator,    we   should   have   had    lo 

m{p  —  r) 

'I 


enclose    the  p  —  r   in    parentheses,  thus: 


But 


when  the  multiplier  is  written  at  the  end  of  the  line,  between 
the  terms  of  the  fraction,  as  above,  it  indicates  that  .he  frac- 
tion, as  a  whole,  is  multiplied  by  ?«.] 

p  —  r 


Subtracting  the  last  product  from  /*,itis  y(J) — (()  —  }u 

Adding  b  to  a,  a  +  b. 

Subtracting  b  from  a,  a  —  b. 

The  product  Q  of  the  sum  and  difference,  {a  -\-  b)  {a  - 
The  product  B  oi'  p  -\-  q  by  p  —  q,  {p  +  O)  {p 

[a  -f  b)  (a 


The  quotient  of  Q  divided  by  Ji, 


b). 

-7). 
■b) 


ij>  +  Q)ip  —  'j) 


*  Tn  matliomatical  ]anp'uaf;e,  n-licn  a  substantive  is  followed  by  a 
symbol  in  this  manner,  the  latter  is  used  as  a  sort  of  proper  name  to 
desifi^natc;  the  substantive,  so  that  the  latter  can  ho  afterward  referred  to 
by  the  letter  without  ambiguity. 

bi  the  i)resent  case,  the  capital  letters  are  used  in  accordance  with 
the  second  general  princii")le,  §  41. 


24 


77//';    LANGUAGh:   OF   ALGKBUA. 


u 


m 


Tlio  fraofion  luivin^  iVfor  its  numerator  und  this  quotient 
lor  iLs  (Icnouiiimtor  i.s 

{nj-  h)  {a  -  h) 

48.  By  tlio  second  general  prineiple,  §  41,  a  sin^jle  sym- 
bol may  be  written  in  place  of  any  al<,'('braie  ex))ressiou  whatever. 
When  several  symbols  indieating  sueh  exi)ressi()ii8  are  eom- 
bined,  the  original  ex'iiressions  nuiy  be  substituted  for  them. 
And  be  treated  m  aeeordauee  with  the  first  prineiple. 


a  —  hx 


EXAMPLES. 

Suppose       P  =  a  -\-  Ix ;  Q 

T  —  X  —  ij\  V  ~  )npq. 

It  is  re([uircd  to  form  the  expression 

PQ  -  TV 
PT-QV' 


The  answer  is 


{a  -f-  bx) 


a 


hx 


m 


—  (^  -  y)  ^npq 


(a  4-  bx)  {x  —  7/)  — 


a 


bx 


■m 


mpq 


2. 

A- 


EXERCIS"!:S 

Form  the  expressions: 
I.     P-T. 
P^Q. 

Vp. 

v(^  -  n 

VK 
VP-QT 


II 


IS- 


IS- 


Q2_  rp2 
(P+    T)(P-T) 

{Q  +  V)(Q-  V)' 
P^-  T 


T  -  P. 
Q-V. 

6.     ViP  -i-  T). 
8.     P^T^. 

lo.     T^V\ 
PJ^ 

{SP-2T)^ 

2(P+  TY 


12. 


14 


1 6. 


{2T-VY 


KXKIii'lSEf^. 


25 


tliis  <|U()Hent 


I  single  syni- 
ioii  wliutover. 
)ns  are  com- 
cd  for  thoiu, 
)!u. 


r 


n 


T) 


2 

^  O 


19. 


/•(r»-r) 


r-(()2-  r)2 


18.      -r. 


20. 


22. 


23 


^*-(c^+na>-r) 


24. 


Qi  —  r» 

__11±Q' 

(r-  r)(r-f  7')' 

F~  (/  +  7' 


EXEnCISES     IN     ALGEBRAIC     LANGUAGE. 


The  fdllowiiip  (jiicstionH  arc  proposed  to  i)nictic('  tlic  Mtiidciit  in  ex- 
pressing  tlu'  rcliitioiis  of  (jiiimtitifH  in  al^a-bniic  luii/^Mta^'f.  Slioiild  any 
of  tljt'iu  otter  diincultifs,  h(!  i.s  reconiinended  to  Hubtstitute  luunhorH  for 
the  al^'-i'hraic  k'ttfrs,  cxamino  the  proccHS  by  wliicli  he  proceeds,  and  then 
Bpl'Iy  tlic  same  process  tt>  tlie  letters  that  he  aj)plicd  to  the  luinihers.  No 
ioliitions  of  equations  are  recpiired. 

1.  TIow  many  cents  arc  tlicrc  in  m  dollars  ? 

2.  How  many  dollars  in  m  cents? 

3.  A  man  had  d  dollars  in  one  ])oekel,  and  h  cents  in  the 
other  ;  how  many  cents  hiid  he  in  all  r     How  many  dollars? 

4.  The  Slim  of  the  ((iiantities  n  and  h  is  to  be  miiUiplied 
by  m.     Express  the  product,  and  its  square. 

5.  A  msin  liaving  h  dollars  p;iid  out  m  dollars  to  one  per- 
son and  }i  dollars  to  another.  Express  what  he  had  left  in 
two  ways  ? 

6.  How  many  chickens  at  k  cents  a  })ieee  can  he  purchased 
for  in  dollars? 

7.  A  man  walked  from  home  a  distance  of  m  miles  at  4 
miles  an  hour,  and  returned  at  the  rate  of  3  miles  an  hour. 
How  long  did  it  take  him  to  go  and  come  ? 

8.  A  man  going  to  market  bought  tomatoes  at  h  cents  per 
peck  and  potatoes  at  k  cents  a  peck,  of  each  an  equal  nund)er. 
They  cost  him  m  cents.     How  many  pecks  of  each  did  he  buy  ? 

9.  How  many  minutes  will  it  require  to  go  a  miles,  at  the 
rate  of  h  miles  an  hour  ? 

10.  A  man  bought  from  his  grocer  a  pounds  of  tea  at  x 
cents  a  pound,  h  pounds  of  sugar  at  y  cents  a  pound,  and  c 
pounds  of  coffee  at  z  cents  a  pound.  How  many  cents  will 
the  whole  amount  to  ?    How  many  dollars  ?    How  many  mills  ? 

11.  A  man  bought  /  pounds  of  Hour  at  m  cents  a  pound, 


26 


THE  ALOEBUAIC   LANGUAGE. 


jiiid  hand'  ;l  tlie  ;,a'ocL'r  an  :r-dollar  bill  to  be  changed?     IlmV 
many  cents  onght  lie  to  receive  in  cliange  ^ 

12.  From  two  cities  a  miles  apart  two  men  started  out  at 
the  same  time  to  meet  each  other,  one  going  m  miles  an  hcur 
and  the  other  ^^  miles  an  hour.  How  h^ng  before  they  will 
meet?  How  far  will  the  first  one  have  gone  ?  How  faV  will 
the  second  one  have  gone  ? 

13.  A  man  left  his  n  children  a  bonds  worth  x  dollars 
each,  and  h  acres  of  land  worth  y  dollars  an  acre  ;  but  he 
owed  })i  uoUars  to  each  of  q  creditors.  What  was  each  child's 
share  of  the  estate  ? 

14.  Two  numbei-s,  x  and  y,  are  to  be  added  together,  their 
sum  multipl'L'd  by  s,  that  product  divided  by  ct  +  b,  and  the 
quotient  subtracted  from  I:.     Express  the  result. 

15.  The  sum  of  the  numbers  p  and  q  is  to  be  divided  by 
the  sum  of  the  numbers  n  and  b,  forming  one  fpiotient.  The 
difference  of  the  numbers  /;  and  q  is  to  be  divided  by  the  dif- 
ference of  the  num))crs  a  and  b,  forming  another  quotient. 
The  sum  of  the  two  (piotients  is  to  be  multiplied  by  r  +  s. 
J^lxpress  the  product. 

16.  The  quotient  of  x  divided  by  a  is  to  be  subtracted 
from  the  quotient  of  y  divided  by  b,  and  the  remainder  multi- 
])lied  i)y  the  sum  of  x  and  //  divided  by  the  dilference  between 
X  and  y.     Express  the  result. 

17.  The  number  x  is  to  be  increased  by  G,  the  sum  is  to  be 
muhii)lied  by  n-{-b,  q  is  to  be  added  to  the  i)roduct,  and  the 
sum  is  to  be  divided  by  r  —  6\     Express  the  result. 

18.  A  family  of  brothers  a  in  number  each  had  a  house 
worth  a  thousand  dollars  each.  What  was  the  total  value  of 
all  the  houses  in  dollars  ?     What  was  it  in  cents  ? 

19.  A  grocer  mixed  a  pounds  of  tea  worth  x  cents  a  pound, 
and  b  pounds  worth  y  cents  a  pound.  How  much  a  pound 
was  the  mixture  worth  ? 

20.  x  +  y  houses  each  had  a  +  b  rooms,  and  each  room 
in  +  n  pieces  of  furniture.  How  many  pieces  of  furniture  were 
there  in  all  ? 

21.  In  a  library  were  p-\-q  volumes,  each  volume  had  p-\-q 
pages,  each  page  p-\-q  words,  and  each  word  on  the  average 
8  letters.  How  many  letters  were  there  in  all  the  books  of  the 
b'brary  ? 

_  22.  A  post-boy  started  out  from  a  station,  travelling  k 
miles  an  hour.  Throe  hours  afterward,  another  one  started 
after  him,  riding  ni  miles  an  hour.     How  far  was  the  first  one 


go 


EXERCISES. 


27 


d  ?     IIow 


icr,  tlicir 
,  uiid  the 

vided  ])y 
lit.  The 
;  the  dif- 
(juotient. 
by  r^s. 

ibtmctcd 

er  muKi- 

bet  ween 

1  is  to  be 
and  tlie 

a  house 
vahie  of 

a  pound, 
a  jwund 

:;h  room 
lire  were 

lad  p-\-q 

average 

ks  of  the 

elb'ng  Jc 
!  st{irted 
first  one 


■ 


ahead  of  the  second  at  tlie  end  of  x  hour.s  after  the  second 

.started? 

23.  Two  men  started  to  make  tlie  same  joiiriiev  of  }ii  miles, 
one  going  r  miles  an  hour,  and  the  other  s  miles  an  hour. 
How  mueli  sooner  will  t!ie  man  going  /•  miles  an  hour  make 
Ids  journey  than  the  one  going  .v  miles  an  hour?  How  much 
sooner  will  the  one  going  ,s'  miles  an  hour  make  his  journey 
than  the  one  going  /•  miles  an  hour  ? 

24.  One  train  runs  from  Boston  to  New  York  in  h  hours, 
at  the  rate  of  u  miles  an  hour.     How  long  will  it  take  another 


tram  running  0  miles  an 


hour  faster  to  nerfo 


tl 


periorm  me  journey 


25.  If  a  man  bought  h  horses  for  /  (hjllars,  and  )i  yoke  of 
oxen  for  m  dolhu's,  how  much  more  did  one  horse  cost  than  one 
voke  of  oxen  ?  How  much  more  did  one  yoke  of  oxen  cost 
than  one  horse  ? 

26.  A  train  making  a  journey  of  'b,v  miles  goes  the  first 
half  of  the  way  at  the  rate  of  /'  miles  an  hour,  and  the  second 
half  at  the  rate  of  x  miles  an  hour.  How  long  did  it  take  it  to 
go  ?     What  was  the  average  speed  for  the  journey  'i 

27.  Two  men,  A  and  B,  started  to  walk  from  Hartford  to 
New  Haven  an(l  back,  the  distance  between  the  two  cities 
being  a  miles.  A  goes  p  miles  an  hour  and  B  q  miles  an  hour. 
How  far  will  A  have  got  on  his  return  journey  when  B  reaches 
Hartford? 

28.  A  man  liaving  k  dollars  bought  h  books  at  ^H  each. 
How  many  books  at  $4  each  can  he  buy  with  the  balance  of 
his  money  ? 

29.  A  man  going  to  his  grocer  with  m  dollars,  bought  .s' 
pounds  of  sugar  at  a  cents  a  pound,  and  r  pounds  of  coll'ee  at 
b  cents  a  pound.  How  many  barrels  of  tlour  at  q  dollars  a 
barrel  can  he  buy  Avith  the  balance  of  his  nujiiey  ? 

30.  A  man  divided  m  dollars  Cijually  among  a  poor  (.'hinese 
and  n  dollars  equally  among  h  orphans.  Two  of  the  C'hinese 
and  three  of  the  orphans  put  their  shares  together  and  bought 
X  Bibles  for  the  heathen.     How  much  did  each  Bii)le  cost  ? 

31.  A  pedestrian  having  agreed  to  walk  the  a  miles  from 
Boston  to  Natick  in  h  hours,  travels  the  first  k  hours  at  the 
rate  of  m  miles  an  hour.  At  what  rate  must  he  travel  the 
remainder  of  the  time? 

32.  A  train  having  to  make  a,  journey  of  ./•  miles  in  h  hours, 
ran  for  k  hours  at  the  rate  of  r  miles  an  hour,  and  then  made 
a  stop  of  m  minutes.  How  fast  must  it  go  during  the  remain- 
der of  its  journey  to  arrive  on  time  ? 


'  \Jv.: 


BOOK    II 


ALGEBRAIC     OPERA  TIONS. 


G  e  11  e  r  a  1   W  e  iii  arks. 

The  algebraic  expressions  formed  in  accordance  with  the 
rules  of  the  preceding  book  admit  of  behig  transformed  and 
sim])lifled  m  a  variety  of  ways.  Tliis  transformation  is  etrccted 
by  operations  which  have  some  resemblance  to  the  arithmetical 
operations  of  addition,  suljtraction,  multiplication,  and  division, 
and  which  are  tiierefore  called  by  the  same  names. 

In  performing  these  algebraic  operations,  the  student  is  not, 
as  in  Aritlimetic,  seeking  for  a  result  which  can  be  written  in 
only  one  way,  but  is  selecting  out  of  a  great  variety  of  forms  of 
expression  some  one  form  which  is  the  simplest  or  the  best  for 
certain  pur[>()ses.  Sometimes  one  form  and  sometimes  another 
is  the  best  for  a  particular  jjroblem.  Hence,  it  is  essential 
that  the  ji^gebraist,  in  studying  an  expression,  should  be  able  to 
see  the  diil'erent  ways  in  which  it  may  be  written. 


Dofiiiitioiis. 

49.  Function.  An  algebraic  expression  containing 
any  symbol  is  called  a  Function  of  the  quantity  repre- 
sented by  that  symbol. 

Ex.  I.  The  expression  Zx^  is  a  function  of  x. 

2.  The  expression  is  a  function  of  x  and   also  a 

n  —  .7' 
function  of  a. 

When    an    expression   contains   several   symbols,  we  may 

select  one  of  them  for  special  consideration,  and  call  the  ex- 

])ression   a  function  of  that  particular  one.    For  instance, 

although  the  expressions, 


DKFrxrrioNs. 


29 


NS\ 


e  with  the 
brined  and 
I  is  oiTocted 
rithmetical 
d  division, 

^ent  is  not, 
written  in 
)f  forms  of 
le  best  for 
es  another 
3  essential 
be  able  to 


)ntaining 
ty  repre- 


id   also  a 

we  may 
11  the  ex- 
instance. 


m  +  nV-^, 
contain   other   symbols   besides   z,   they   are   both   functions 
of  2\ 

i)0.  An  Entire  Function  is  one  in  which  the  quan- 
tity is  nsed  only  in  the  operations  of  addition,  subtrac- 
tion and  multiplication. 
Example.     The  expressions 

ax  +  >/, 
{cfi  —  if)  a-3  —  (//■•^  -\-  y)^^  —  ^  +  (h 
are  entire  functions  of  x.     But  tlie  expressions 

ax  -\-  y         1     o   / 

—-     and     3v^ 

ax  —  y 

arc  not  entire  functions  of  .r,  because  in  the  one  x  appears  as 
part  of  a  divisor,  and  in  the  other  its  square  root  is  extracted. 
An  entire  function  of  x  can  always  be  expressed  as  a  sum 
of  terms,  arranged  according  to  the  powers  of  x  which  tliey 
contain  as  factors.     The  form  of  the  expression  will  then  be 

A  +  Bx  +  Cx^  +  /):r3  +  E^c^  +  etc., 
where  A,  B,  C,  etc.,  may  represent  any  algebraic  expressions 
wliich  do  not  contain  x. 

51.  Like  Terms  are  those  which  are  formed  of  the 
same  algebraic  symbols,  combined  in  the  same  way, 
and  difier  only  in  their  numerical  coefficients. 

Ex.     The  terms  ax,  2ax,  —6ax  are  lilvc  terms. 

5*^.  The  Degree  of  any  term  is  the  number  of  its 
litend  factors. 

Examples.  The  expression  abxy  is  of  the  fourth  degree, 
l^ecausc  it  contains  four  literal  factors. 

The  expression  .r^  is  of  the  third  degree,  ])ecausc  the  letter 
X  is  tal^en  three  times  as  a  factor. 

The  expression  ah^x^  is  of  tlie  sixth  degree,  l)ecause  it  con- 
tains a  once,  b  twice,  and  x  three  times  as  a  factor. 

When  an  expression  consists  of  several  terms,  its 
degree  is  that  of  its  highest  term. 


30 


.1 L G EBRAIG   OPERA  TI0N8. 


CHAPTER       I. 

ALGEBRAIC    ADDITION    AND    SUBTRACTION. 


Algebraic  Addition. 

5I>.  By  the  language  of  Algebra,  the  sum  of  any  number 
of  ({uiuitities,  positive  or  negative,  may  be  expressed  by  writing 
them  in  a  row,  with  the  sign  +  before  all  the  jiositive  quan- 
tities, and  the  sign  —  before  the  negative  ones. 

Ex.  A.-\-B—D—X'\-Y,(itQ.,  is  the  algebraic  sum  of  the 
several  (quantities  A,  B,  —D,  —X,  Y,  etc. 

54.  To  siiiiplify  an  e.vpressioji  of  the  siun  of  several 
quantities. 

1.  AYhen  dissimilar  toiins  are  to  be  added,  no  sim- 
plification can  be  effected. 

Ex.  If  we  require  the  sum  of  the  five  expressions,  a,  — xi/, 
mp,  nq,  and  —hhs,  we  can  only  write, 

a  —  xy  -\-  mp  -\-  7iq  —  hhs, 

according  to  the  language  of  Algebra,  and  cannot  reduce  the 
expression  to  a  simpler  form. 

2.  If  mere  numbers  are  among  the  quantities  to  be 
added,  their  algebraic  sum  may  be  formed. 

Ex.  The  sum  of  the  five  quantities  —8,  ah,  5,  mnj),  —15, 
is  found  to  be  —  18  -f  ah  -\-  mnp. 

8.  When  several  terms  are  similar,  add  the  coeffi- 
cients and  affix  the  common  symbol  to  the  sum. 

When  no  numerical  coefiicicnfc  is  written,  the  coefficient 
f  1  or  —1  is  understood.     (§  44.) 

EXAMPLES. 

a  +  a  =:  '^a  [because  1  -f  1  —  2]. 
3rt  —  a  =  a  [because  2  —  1  :=  1]. 


4 


ALGEBRAIC   ADDITION. 


m 


TION. 


any  numl)cr 
(1  by  writing 
jsitive  qiiiin- 

}  Slim  of  the 
of  several 
ed,  no  sim- 

3ns,  a,  —xy, 

:  reduce  the 
ities  to  be 
mnj),  —15, 

the  coefR- 
im. 

i  coeflBcient 


4 


3^  4.  4rt  _  7a  =  0  [because  3  +  4-7-0], 

a  +  -ix  —  'da  —  hx  =  —2a — 3x  [adding  the  <('ti  and  the  i-'s]. 

—  'Saxi/  +  -ihii  —  2axij  -\-  hm  z=z  —  haxij  +  hhm. 

Add  tlie  expressi<uis, 

I.     ;./•  +  bbif,  2x  —  obip,  —  4:X  —  bbtf,  bx  —  hif,  x  —  bif. 


WORK. 

7.K  +  r)/>?/2 


5^' 


For  convenience,  the  several  t(Tnis  may  bo 
written  under  eacli  other,  as  in  the  margin.  The 
r-oeliicients  of  x  are  7,  3,  —4,  5,  and  1,  of  which 
the  algebraic  sum  is  11.  The  coefficients  of  2/'^ 
are  5,  —3,  —5,  —1,  —  1  ;  the  sum  is  —5.  Hence 
the  result. 

Sumy  l\x  —  bbif 

2.     ^ax  —  y  —  2y  -{-  5,  7ax—y  —  9-\-a)>i,  2ax—y  —  '3-{-bj). 

Here  2.7",  am,  and  p, 


2x  —  'Aby'^ 

4:X   —   6  b  I/' 

x-    b\f 


all  being  different  sym- 
bols, the  terms  contain- 
ing them  do  not  admit 
of  simplification  (§  54, 
1).  The  numbers  5, 
—9.  —3,  are  added  by 


WORK. 

8(7.^2  —    y  —  2x  -\-  b 

"t'ax'^  —   y  —  0  +  (17)1 

(fX^  — 


1/ 


•» 
0 


+  5;j 


Sum,    —  ']y  —  2x  —  7  -{■  am  +  bp 

the  rule  (§  54,  3).    The  coefficients  of  ax"^  cancel  each  other  (8—7—1  =  0). 
3.  Add  G  (a:  +  y\  5  {x  +  y)  +  a,  2  {x  +  //)  -  3a. 


Here  the  aggregate,  x  +  y,  enclosed  in 
parentheses,  is  treated  as  a  simple  symbol. 

Note.  When  the  student  can  add 
the  coefficients  mentally,  it  is  not  neces- 
sary to  write  the  expressions  under  each 
other.  Nor  is* it  necessary  to  repeat  the 
symbol  after  each  coefficient. 


WORK. 

c  {x  4-  y) 

5  H 

3  -3 


a 


Sum,  \3  {x  -{-  y)  —  2a 


EXERCISES, 


1.  3a  +  7^*  —  8r  +  d,  3a  —  2b~{-c  —  o,  —a  —  b  —  c—d. 

2.  7rt  -  {x  +  y),  8«  -  {X  +  y),  3  (.^•  +  y)  -  IGrt. 


3- 


-j'2  _  2x  —  b,  2x'^  —  3x  +  8,   —  9.r2  +  bx  +  3. 


4.  .r^  +  2x  —  y,  4.r-^  +  7x  —  2y,  —  2j^  -\-x  —  9y,    —  3x^ 

—  X  —  y. 

5.  9  {a  +  bf,  10  {a  +  bf,  {a  +  bf,  2  {a-\-bf,   -x-y-::. 

6.  2  {m  +  n)  +  3  («  +  b),     {a  i- b)  —  {in  +  11),     {a  +  //) 

—  (ni  +  n). 


7.     '^a^-.2a^  -{-3ax, 


(7,3  —  f(2 


ax,  —  Cut^  +  3a"-  —  2ax. 


'62 


ALG EliUAlC    OPEUA  TJOJV>S. 


{m  -\-  ny  —  y. 

10.  (kt  {x  —  i/),  i)a{.^—i/),  2a  {x— I/),  a{x — i/). 

11.  2  (in  --  u)  X  -\-  'Z,     3  [in  +  ?i)  x'  --  5,     5  (//i  +  n)x  ■—  (i, 

y  y     ^^  ^     1 


12.     3";',  3''  + 3-/,  '^ 
a       a         0    a 


X      in     ^  X 

13.     ,  2 

^      y      n'     y 


2^,3^ 


^m       X         m 

3  — ,  4 4  — ■ 

?i         ?/  /i 


rw  -f-  n        m  +  ?i        wi  +  w         ?/i  +  ^a 

15.  Of  two  fjirincrs,  tlio  first  had  2x  —  3y  acres,  and  the 
second  liad  x  —  y  acres  more  than  the  first,  llow  many  acres 
luul  they  botli? 

16.  A  had  2x  dollars,  B  had  ?/  dollars  less  than  A,  and  C 
had  2?/  dollars  more  than  A  and  iJ  togecher.  How  many  had 
they  all  ? 

17.  A  father  gave  his  eldest  son  x  dollars,  his  second  5  dol- 
lars less  tlian  the  first,  his  third  5  dollars  less  than  his  second, 
and  his  fourth  5  dollars  less  than  his  tiiird.  How  much  did 
he  give  them  all  ? 

55.  Addition  with  Literal  CQefficients.  When  dif- 
ferent terms  contain  the  same  symbol,  multiplied  by 
different  literal  coefficients,  these  coefficients  may  be 
added  and  the  common  symbol  be  affixed  to  their 
aggregate. 

EXAMPLES. 

1.  As  we  reduce  the  polynomial 

i)X  -\-  5,<;  —  2x 
to  the  single  term      (G  +  5  —  2)  x  =.  3a', 
so  we  may  reduce  the  polynomial 

ax  ■\-  hx  —  ex 
to  the  single  term,  {a  ^h  —  c)x. 

2.  Tiie  expression 

mx  4-  ny  —  hx  ■\-  dy  ■{■  a  -\-  h 
may  be  expressed  in  the  form 

{ni  —  b)  X  +  («  -I-  (/)  y  -f-  a  -f-  h» 


\)VV. 


fi 


SUBTRACT  ION. 


33 


\-y)- 


n 


3rcs,  and  the 
V  niaiiy  acres 

an  A,  and  C 
w  many  had 

econd  5  dol- 
I  his  second, 
3W  much  did 

When  clif- 
Itiplied  by 
ts  may  be 
d  to  their 


EXERCISES. 

(A)lloct  the  coefficients  of  x  and  y  in  tiie  following  ex- 
pressions: 

1 .  ax  +  1)1/  -\-  mx  +  «//. 

2.  uuix  +  'Zbij  -\-  p([x  —  \hy. 

3.  ;3./;  —  9^1  +  i')bx  —  -i//  4-  '\ax  +  m  -f  n. 

4.  8r/y  +  8^.<'  +  ^v/  +  7^'  —  5^  +  :c  —  5^. 

5.  ax  -\-  djj  -{■  cz  —  w:c  —  711/  —  pz. 

6.  2dx  +  3f^  4-  4/2  —  2fx  —  'Sdy  +  -iez. 

7.  j^  r/^  — -^o;  4- ^  ^i/ +  Oa:c. 

8.  2ax  —  /;?/  —  'Sbx  —  4^^. 

1  2,         1  3 

9.  ,^ax  +  ,_^bi/  - -mx -\-  ^ny. 

2  1 

10.     ^mx  -\-2y  —  '6ax  —  (jcx  4-  ay  —  .  ?//.r  4-     <Z,c. 

.1       II.     babx  —  3^/i?i?/  —  abx  4-  4c^/7/  —  ^/x*. 
12.     3f/^  4-  2bx  —  (Ix  4-  2ay  —  '6bx. 


I3. 

14. 

15- 
16. 

17- 


1 


ay  —  3.r  4-  -  ?/ 


ay  _  5^  4-  2/. 


3;?z:i  —  «.-«  —  -  ay  4-  .1:  4-  dx  —  y. 

'^abx  —  my  4-  26•A/.^'  —  f///  4-  '^x. 
bmVy  —  6x  -{■  -^Vy  —  dVx  —  y  4-  V;?/. 

4^/2;  —  G^  4-  rtV^  4-  ex  —  Vy  —  ■^ctVy  4-  Vx. 


Alj?ebr?iic  Subtraction. 

,10.  Def.  Algebraic  Subtraction  consists  in  ex- 
pressing tlie  difference  of  two  algebraic  quantities. 

Hide  of  Subtraction.  It  has  been  shown  (§  21)  that 
to  subtract  a  positive  quantity,  5,  is  the  same  as  to 
add,  algebraically,  the  negative  quantity,  —h.  Also, 
'hat  to  subtract  —h  is  equivalent  to  adding  4-^.  Hence 
the  I'ule : 

CJimidp  the  algehvaic  sign  of  all  the  terms  of  the 
suhtrahend,  or  conceive  them  to  he  changed,  and  then 
proceed  as  in  addition. 


'M 


A LOEBRA W   OPERA  TIONS. 


NUMERICAL  EXAMPLES. 

Min,,    10  +  C==l(j    10+   G==1G  10+   (5  =  10     10+  C,  = 

Sill)!.,     0       =   \)       0—  4=  5  _[>—  8=   1       9  —  12  = 

Kfiii.,     1+0=   T       1+10  =  11  1  +  1-4  =  15        '~^" 


k; 


1  +  18=     111 


ALGEBRAIC       EXERCISES. 


I.  From 
Subtract 


ox  —  hnj  +  ,")/>  +  c, 
X  —  ^lUij  —  8Z»  +  cl. 

WORK. 

Minuend,  Zx  —  ^ay  +    5Z»  +  c 

Subtrahend  with  signs  clianged,     —  a;  +  '^ay  +    Sb  —  d 

Difference, 


2x  +  'Say  +  13b  -{-  c  —  d 

Next  we  may  simply  imagine  the  signs  changed. 
2.     From        Ix  —    4  J:?-//  —  VZcy  +    8^  +  3«c 


Take 


2.r  +    7^.i-?/  +    8r;?/  —    5b  — 2d 


Diff.,   5^  —  Ubxy  —  20cy  +  13/^  +  3ac  +  2(/ 


From 
Take 

From 
Take 


Sa  +  9^  —  12<?  —  18^?  —  4^  +  3*:?^/ 
Wa  —  7b—    8c  —  25r7  +  3x  —  4y 

257;?  +  201:22  ^  92^  +  35^^a:  —    G 
140;2  —    82z^  +  20?/  +  d2f(X  +  14 


5.  From  8a  +  14^  subtract  6a  +  20b. 

6.  From  a  —  ^  +  6-  —  d  take  —  a  +  Z>  —  c  +  r/. 

7.  From  8a  —  25  +  3c  subtract  4a  —  GZ*  —  c  —  2^7. 

8.  From  22-2  —  8,r  —  1  subtract  5x^  —  6x  +  3. 

9.  From  4:X^—'Sa^  —  2x^  —  7x  +  9  subtract  a;^— 2a;3 
+  Hx  -  9. 

10.  From  2:^  —  2a.i'  +  3a^  subtract  x?  —  ax  +  a\ 

11.  From  a3  —  3a^  +  SaZ'^  —  b^  subtract  —  a^  +  Sa^T*. 

12.  From  ;,v3  —  2.r2  +  22'  +  2  subtract  4^:3.  ^22-2— 2a;— 14. 

13.  From  5{x  —  y)  +  7  (a;  —  2;)  +  9  [z—x)  take   9  (a;  —  ;?/) 
+  7  (.'K  —  ^)  +  5  (/:  —  .r). 

14.  From  12  (a  —  Z^)  -  3  (a  +  ^)  +  7a  —  2J  take  7  (a— Z») 
—  5  (a  +  Z>). 


15.   From 


X 

II 


11 


// 


15     take 


X 


5-^  +  C-" 


iC  Z* 


f 


I        i 


BUBTRACTION, 


35 


0+   fi  — 
\)  —  VZ  — 


1  +  18  = 


lu 


U  +  c 

(1. 


)  +  'Zd 


-d 


y 


d. 

—  2d. 

(A 

+  3a^. 
r2—2x—U. 
B   9  {x  —  y) 

ke  '){a-b) 


X  0 


Clejiriiij^  of  Par<»iithese.s. 

57.  In  g  4-.',  :2,  it  WHS  shown  that  an  iiLTC^rc^iilc  of  tonus  iii- 
(•liidt'd  bi'twecii  jtaronlliC'SC'S  might  Ix'  addi-d  or  .suhtrai'tod  by 
giuipiv  writing  +  or  —  hot'oiv  tiic  iiui'untht'SfS. 

When  an  aggregate  not  multiplied  by  a  factor  is  to  be  added 
or  subtracted,  the  paror. theses  may  be  removed  by  the  rules 
lor  addition  and  subtraetioUj  as  follows: 

5S.  Flits  Sign  before  Parentheses.  If  tlio  paren- 
theses are  preceded  by  tlu;  sign  +,  tliey  may  bo 
removed,  and  all  the  terms  added  without  change. 

Example  i.    27  + (8— 5-4  +  7)  -27  +  8-5-4  +  7  =  33. 

2.  VI  +  {a  ~  X  —  y  -\-  7^  —  m  -\-  a  —  x  —  y  -^  z. 

3.  2x  +  (-  3^;  -  5//)  +  (3//-4^0  +         -2//) 

=  2x  —  3a;  —  5y  +  3^  —  4rt  +  2y  —  2a 
=  —  X  —  (ja. 

The  sign  +  which  precedes  the  parcnlhosos  should  also  be 
considered  as  removed,  but  if  the  first  term  within  the  ])arcn- 
thesis  has  no  sign,  the  sign  +  is  understood,  and  must  be 
written  after  removing  the  parentheses. 

EXERCISES. 

Clear  of  parentheses  and  simplify 

1.  x  —  y-{-  {x  +  y). 

2.  a:-}-  y  +  (y  -  x). 

3.  3rt&  —  2mp  +  {ab  —  ox  —  2mp). 

4.  2ax  —  3by  +  {inx  —  2ax  —  j}z  +  V)y). 


5. 


='^(^^^)+(^^^)• 


59.  Minus  Sign  before  Parentlieses.  If  the  paren- 
theses are  preceded  by  the  sign  — ,  they  may  ])e 
removed  and  the  algebraic  sign  of  each  of  th(^  inchided 
terms  changed,  according  to  tlie  rule  for  subtraction  in 


%m. 


I. 


E  X  A  M  P  I-  E  S  . 


that  is,  27  —  6  =  21. 


(8 -5-4  + 7)  =  27 -8  +  5  +  4-7 


86 


ALGEBRAIC   OPERATIONS. 


i        1 


2.  m  —  {—  a  —  })  -\-  y  -\-  o-)  ■=.  m  -\-  a  •{-  p  —  y  ^  oc. 

3.  '^a  +  ./;  —  (:irt  —  ox)  —  (:)./•  —  a)  —  '•Sa  +  x  —  '^a  -\-  5/ 
—  'Jx  +  a. 

JSiinpIifying  u.s  in  §  .Jir,  this  rt'duces  to  ^a  —  ',ix. 

E  X  E  F<  C  I  S  E  S. 

Clear  the  following  cxpres.sions  of  ptironthcsos  and  reduce 
1 1)0  resulLs  to  the  simplest  form  by  the  method  of  §  a-i. 

1.  ab  —  {7)1  —  3ab  +  2ax)  —  "tad. 

2.  X  —  {a  —  x)  -\-  (x  —  a). 

3.  2b  -\-  {b  -  :lc)  ~{b  -{-  2r). 

4.  ix  —  ;}//  +  2z  —  ( -  7x  +  5^  —  3z)  —  {x  —  tj), 

5.  ''iitx  ~  21)!!  —  {'(^((x  ■{-  Wbji)  —  {^ax  —  '6b y), 
C.  {a  —  x)  —  {if  4-  .?')  +  ;i.''. 

7.  —  (re  —  /y)  —  (/;  —  (')  —  {r  —  a). 

8.  —  (oy/i  +  2t/)  —  (.'Jyy?  —  :i;y)  +  iWi. 

(U).  We  may  reverse  (he  process  of  clearing  of  parentheses 
by  collecting  several  terms  into  a  single  aggregate,  and  chang- 
ing their  signs  when  we  wish  the  parentheses  to  be  preceded 
by  the  minus  sign,  'riic  proof  of  the  operation  is  to  clear  the 
parentheses  introduced,  and  thus  obtain  the  original  expression. 

EXERCISES. 

Eeduce  the  following  expressions  to  the  form 

X  —  {an  ayyrcyate). 

1.  X  —  a  —  b.  Ans.  x  —  {a  -\-  b). 

2.  X  —  m.  —  n. 

3.  a  +  X  —  3x  -\-  2y.  Ans.  x  —  {—  a  -\-  3x  —  2y). 

4.  —  3b  -It  X  -^  2c  -\-  bd. 

5.  2x  —  2a  +  2b.  Ans.  x  —  {—  x  -i-  2a  —  2b). 

6.  2x  -i  a  —  b. 

7.  3x  —  2m  +  2n. 

8.  3x  -{■  ab  —  m  —  3ab  -\-  2m. 

9.  X  —  2m  —  {3a  —  2b).       Ans.  x 

10.  X  -\-  3  —  {a  -j-  b). 

11.  X  '\-  a  —  {b  —  c)  -f-  {m  —  n). 

12.  X  —  {am  -\-  b)  —  {p  —  q)  —  {am  —  n). 


{2m  -f  3«  —  2b), 


13- 


X  —  {a  +  b)  —  {])  —  q)  —  {ni  —  n). 


iSUDTBACTION, 


37 


X. 


^s  and  reduco 
t'  §  54. 


-y)' 


). 


jf  parontliGses 
0,  and  L'lian<i- 
)  be  preccdcMl 
is  to  clear  the 
lal  expression. 


-  3a:  —  )ly). 
2a  —  2b). 


-  3a  —  2b). 


Conipouiid  Pare  111  hcses. 

r>l.  When  parentheses  of  achlition  or  subtraction  nvr  cn- 
josid  lietNveen  others,  they  may  l)c  si'parati-lj  reniowd  \>\  tli.- 
)reci'illn^^  rules. 

We  may  either  bc^in  witii  tlie  outer  ones  and  go  inward, 
or  lu'uin  witii  the  inner  onus  and  <(o  outward, 
it  is  connnon  to  begin  with  the  inner  ones. 

EXAMPLES. 

Clear  of  parenilieses: 

1.  f-b-\d-[r,-{b-a)]\l 

Beginning  witli  the  inner  parentheses,  the  expression  takes, 
in  succession,  the  following  forms : 

f-\e-\d-[c~b-{-a]\] 

=  /-]'<'-  1'/-^'  +  ^-^'}] 
=  f  -  \''  -d  +  c  -  b  -^  (('] 
=  f  —  e  -\-  d  —  c  +  0  —  a. 

2.  X  —  [—  {a  -}-  b)  -h  {m  +  n)  —  (.6-  —  ?/)]. 
Eemoving  the  inner  parentheses,  one  by  one,  we  have, 

X  —  [—  fi  —  b  ~\-  m  -}-  n  —  X  -\-  y] 

=  X  +  a  -\-  b  —  7)1  —  n  -{-  X  —  y. 


EXERCISES, 

Eeniove  the  parentheses  in  the  following  expressions,  and 
combine  terms  containing  x  and  y,  as  in  §§  54  and  55 

1.  m  +  [-  (p  -  q)  -t-  (rt  -  Z.)  -j-  (-  c  4-  ^0]- 

2.  m  —  \—{a  —  b)  —  (2)  +  q)  +  (u  —  k)\. 

3.  Hax  —  [{2ax  -\-  by)  —  {3ax  —  by)  +  (—  7ax  +  2by)]. 

4.  a  —  {a  —  \a  —  [a  —  {a  —  a)]  \]. 

5.  p  —  [a—  b—  {s  -{-  t  -i-  a)  -\-  {-  m  —  71)]. 

6.  2ax  —  [3((x  —  by  —  {^iax  -|-  2by)  —  {bax  —  3by)\ 

7.  ax-\-by-\  cz  +  {2ax—3cz  —  {;2vz-]-bax)  —  {7by—dcz)]. 

8.  X—  \  2x  —  y—  [3x  —  2y  —  {4x  —  3y)]  \. 

9.  ax  —  hz  —  \ ax  -{■  bz  —  \^ax  —  bz  —  [ax  +  bz)]\. 
10.  my  —  \x  +  3y  +  [2)ny  —  3  {x  —  y)  —  4rti]  +  5 [. 


38 


ALaji'JiliAJC  OPKUA  noNs. 


11.  ax  -I-  icx  —  {inx  -f  ex  —  y)  -h  [fn-^'  —  ((^x  +  y)]. 

1 2.  '>i((X  —  '>\0x  —  (  —  .hit/  —  Ihtz  '\-  ^hij)  —  Whz. 

13.  i:J^/.r  4-  -'7/  —  d  —  [^:(«t  f  ('/y  ^-  '/)]  —  4ry. 

14.  m  +  4.r  —  [—  4//  -(-  :i.«  +  (<^//  —  ./•)  +  ))]. 

15.  :iaV.y  —  3//i  —  [^Va;  —  G/i  -)-  (V//  —  ^a///)]. 


-♦♦♦- 


:,  :)!• 


I 


CHAPTER    II. 

M  ULTI  PLICATION. 

03.  The  product  of  several  factors  can  always  Im^ 
ex])ressed  by  writing  them  after  each  other,  and  enclos- 
ing those  which  are  aggregate's  within  parentheses. 


lac 


EXAMPLES. 


The  product  oi  a  -\-  h  hy  c  =1  c  {a  -\-  b). 
The  product  of  — ~  by  x  —  y  =  {.r 


y) 


X  -\-  y 


The  product  of  n  j-  b  by  c  -\-  d  =  {c  -\-  d)  {a  +  b). 
Such  products  may  be  transformed  and  sinipHfied  by  tho 
oiK'ration  of  algebraic  muhipHcation. 

General  Laws  of  Multiplication. 

03.  Law  of  Commutation.  Multiplier  and  multi- 
plicand may  be  interchanged  without  altering  the 
product. 

This  law  is  proved  for  whole  numbers  in  the  following  way. 
Form  several  rows  of  quantities,  each  represented  by  thy 
letter  «,  with  an  equal  number  in  each  row,  thus, 

a  a  a  a  a  a 

a  a  a  a  a  a 

a  a  a  a  a  a 

a  a  a  a  a  a 

a  a  a  a  a  a 


J 


■I 


7.r  +  y)]. 


I  always  bo 
and  cncloH- 
[itheses. 


+  :'/ 

— — —  • 

«  + J). 

plified  by  the 


and  multi- 
Itering   tli(^ 


:)llowing  way. 
nted   by   the 


MULTIPLICATION. 


39 


I 

V       Let  m  Ikj  tho  minilKT  of  rows,  and  n  llie  huhiIkt  ot  (i\  in 
cacli  row.     TIkii.  counting  by  rows  there  will  bo 

m  X  n  (juuntitiud. 

Counting  l)y  columns,  there  will  be 

n  X  »ii  (|uuntitics. 
Til  en '{'ore,  tn  x  n  =  )t  x  tn, 

-,!•  nm  ■=.  7nn. 

'A      <U.  Lf'W  of  Association.     Wlicn  tlioro  are  tlirce 
factors,  w,  //,  and  a, 

///  {ua)  =  {mn)n. 

Example.        3  x  (5  x  8)  =  3  x  40  =  120. 
(3xr))x8  =  15  +  8  =:  im 

Proof  f 07'  Wliolc  yinnhcrs.  li  a  in  the  {il)ovc  schomo 
represents  u  number,  tlie  sum  of  eacli  row  will  be  riK.  Because 
there  are  ;;/  rows,  the  whole  sum  will  ))e  ///  (;/^/)- 

But  the  whole  number  of  r«'s  is  iniL     Therefore, 

7n  [na)  =  {vni)  a. 

(>;■;.  The  Dislrihutiiy'  ,'jaio.  Tho  i)rodnct  of  an  ac;- 
gr(^<iat(^  by  a  factor  is  equal  to  the  sum  of  the  ])r()(lu('ts 
of  each  of  the  parts  which  form  the  aggregate,  by  the 
same  factor.     That  is, 

|H  t^ip  -\-  q  -\-  r)  =  mj>  4-  }nq  +  wr.  (1) 

Proof  for  ]\lioIe  Numhcrs.  Let  us  write  each  of  the  (juan- 
tities  ]),  q,  r,  etc.,  m  times  in  a  horizontal  line,  thus, 

2^  +  P  -\-  P  +  etc.,  m  times  =  mp. 

■  mSk                       q  +  q  +  q  -\-  etc.,  m  times  =  mq. 

r  -\-  r  -{-  r  -\-  etc.,  m  times  ==  mr. 

etc.  etc.           etc. 

If  we  add  up  each  vertical  column  on  the  left-hand  side, 
the  sum  of  each  will  be  2^  -{■  q  +  r  ■{■  etc.,  the  columns  being 
all  alike. 

Therefore  the  sum  of  the  m  columns,  or  of  all  the  quanti- 


ties, will  be 


wi  (j5  -f  (7  +  r,  etc.). 


40 


ALGEBRAIC  OPERATIONS. 


i'  , 


P   (" 


m 


\  \ 


m 


The  first  horizontal  line  of  p'B  being  mp,  the  second  mq, 
etc.,  the  sum  of  the  right-hand  column  will  be 

mp  +  7nq  +  mr,  etc. 

Since  these  two  expressions  are  the  sums  of  the  same  quan- 
Lities,  they  are  equal,  as  asserted  in  the  ecjuation  (1). 

Multiplication  of  Positive  Mononiijils. 

GO.  Rule  of  Exponents.     Let  us  form  the  product 

a:"'  X  x\ 

By  §  37,     X'''  means  xxx,  etc.,  taken  m  times  as  factor. 
X''  means  xxx,  etc.,  taken  n  times  as  factor. 

The  product  is  xxxxx,  etc.,  taken  {m-\-n)  times  as  factor. 

TJierefore,  .^•"'  x  x"  —  x"'+\ 

Hence, 

Theorem.  The  exponent  of  tlie  product  of  like  s^.i 
bols  is  tlie  sum  of  the  exponents  of  the  factors. 

07.  As  a  result  of  the  laws  of  com  mutation  and 
association,  the  factors  of  a  product  may  be  arranged 
and  multiplied  in  such  order  as  will  give  the  product 
the  simplest  form. 

(>S.  Any  product  of  monomials  may  be  formed  by 
combining  these  principles. 

ExAMi'LE.     Multiply  ^m,n^x^\f  by  Ihnx^y. 

By  the  rules  of  algebraic  languag:,  the  product  maybe  put 

into  the  form 

bmn^x^  ij'^lhnx^y. 

By  interchanging  the  foctors  so  as  to  bring  identical  sym- 
bols together, 

5  •  7  Z»  m  n^  n  a9  x^  if  y. 

Mijltiply  ing  the  numerical  factors  and  adding  the  exponents, 

the  product  becomes 

^blmrt^ofiy^. 


/^" 


1. 

2. 

4- 

6. 

8. 

TO. 
I  I. 

12. 
14. 


^"^IP 


M  UL  TIP  Lie  A  TION. 


41 


le  second  mq. 


lie  same  quan- 

niials. 

loduct 

as  factor. 
IS  factor. 

es  us  factor. 


of  like  S3/.1 

ors. 

utation  and 
be  arranged 
the  product 

formed  by 


!t  maybe  put 


lentical  svm- 


le  exponents, 


Of).  Wc  tlms  derive  the  following 

l^Li:.  Miilliplij  the  unDicricnl  cnnjficicnts  of  the 
fttrf (>/'.<,  (ifp-y  f(U  the  litei'dl  parts  of  the  factors,  and  ^ive 
ii)  r/'ch  the  sicni  of  its  exponents  in  the  separate  factors. 

EXERCISES. 

1.  .Aliiltijil;,'  xij  by  x'^ip 

2.  ]\inltii)ly  3ra'  by  'iahx'^.  3. 
4.  ^\\\\\i\Ay  "llmji  hy 'iifim.  5. 
6.  Multiiily  hx'ifz  by  x'-ifz.  7. 
8.  ]\Iiiliiply  'iahni  by  "2iuha.  9. 

10.  .Mu]ti[)ly  2-(j)Hj)fjr  by  .l-Cjpqrs. 

11.  ]\Iultii)ly  l'2((.ri/  by  Vlxijz. 


j'^if. 


Ans. 

?vrultiply  hm'^y  by  '^m^x. 
Multijtly  'Zcwi  by  2ma. 
Multiply  ^xyz  ])y  .'3.r//,?. 
Multiply  '^ah\fi  by  ia3\i\ 


3 


13.  Multiply  '-.n'^Jc  by  •i;;?^*. 


12.  Multiply  [  /^i^a.-^  by  "//i^//'. 

/i  O  1. 

7 
14.  Multiply  ^:^abcd  l»y  4rZ^/^. 

70.  When  we  have  to  find  the  product  of  tlirec  or  more 
(|uantities,  we  multiply  two  of  them,  then  that  product  by  the 
third,  that  product  anain  Ijy  the  fourth,  and  so  on. 

Ex.     2((b  X  2a^  x  3«^*2  x  3bmx7j  —  SOa^b^nixt/. 

ExEKcisES.     Multiply 

15.  urxxmij  xinz.  16.     axxbxxcxxdx. 

1 7.  oc'^m  X  ■ib'n  X  uni.  18.     ab  x  2bc  x  7ca. 

19.  'ilmn^  X  Unj)^  x  9pnr*: 

20.  (lb  X  ac  X  ad  x  (nn'i^  X  y  X  2yz  x  zx. 

2 1 .  aiiix  X  anx  x  amxy  x  anxy  x  anixyz. 

22.  ah:  X  a^y  x  ax^  x  ay^  x  «V  x  a'y"^  x  xhf. 

23.  2am  X  'dan  x  a^  x  vi^  x  ^mx  x  2hx. 

Rule  of  Signs  in  Multii)li<ation. 

71.  It  was  shown  in  §  25  that  a  product  of  two  factors  is 
])()sirive  when  the  factors  have  like  signs,  and  negative  when 
they  have  unlike  signs.     Hence  the  rule  of  signs, 

+    X   +     makes     4-> 
+    X  -        ''         -, 

-  X   +         "  -, 

—  X  —        "         +. 


'■'% 


42 


ALGEBRAIC  OPERATIONS. 


•ill 


Examples.    The  quantity  a 


w^ 

Multiplied 

^>y 

3 

makes 

+  Sa. 

...     '    1 

a 

2 

u 

+  2a. 

h  /  " 

i( 

1 

(( 

+    a. 

K    .'  1 

i( 

0 

'i 

0. 

ji''  ■.■'"*■  ^ 

ti 

-1 

u 

—    a. 

.'f 

a 

-2 

a 

—  2a. 

The 

quantity  —  a 

Multiplied  by 

3 

makes 

-3a. 

a 

2 

a 

—  2a. 

if 

u 

1 

(I 

—   a. 

r* 

a 

0 

it 

0. 

I 

t( 

-1 

u 

-f-    a. 

u 

-2 

li 

+  2a. 

73.  Geomein'cal  lUustraHon  of  the  Ride  ofSif/ns.    Suppose 

tlie  quantity  a  to  represent  a  length  of  one  eentimetre  from 

the  zero  point  toward  the  right  on  the  scale  of  §  11. 

Then  Ave  shall  have 

0 

a  =  this  line    |  | 

The  product  of  the  line  by  the  factors  from  -\-3  to  —3 
will  be 


a  X  3, 

«  X  2, 

a  X  1, 

a  X  0, 

«  X  —  1, 

ax  —2, 

ax—  3,      I 

We  shall  also  have 


0 

0 


0 


I 

0 


i 


—  a  =  this  line 


M  UL  TIP  Lie  A  TION. 


43 


The  proilncts  by  the  same  factors  will  be 


'IS.  Suppose 
timetre  from 
1. 


_(-3  to  —3 


0 

0 


—  rt  X  'J,       I  1  I 

—  rt  X  2,  I  I 

—  a  x^•,  i""" 

—  rt  X  0, 
— «  X  —  1,  I  I 

0 

—  rt  X   —  3,  I  I  I 

0 

—  ax  —  3,  ill! 

These  results  are  embodied  in  the  following  two  theorems  : 

1.  Multiplying  a  magnitude  by  a  negative  factor, 
multiplies  it  by  the  factor  and  turns  it  in  the  opposite 
direction. 

2.  Multiplying  by  —1  turns  it  in  the  opposite  direc- 
tion without  altering  its  length. 

Note.  When  more  than  two  factors  enter  a  product,,  the 
sign  may  be  determined  by  the  theorem,  §  2G. 


I 

3 

4 

5 
6 

7 
8 

9 

10 

II 

12 

13 
14 
15 


EXERCISES, 

am  X  ab  X  ac  X  ad.         2.     ax  x  —hxxcxx  dx. 
X  X  —ax  X  —ahx  x  —obex. 
?>ax  X  —'^aW  X  —  bahnx. 

—  7 11)^1/  X  — M^y^  X  hax. 

—  2nzn  X  —5n^x"'  x  —n^yz  —  xPK 
2m  X  n  X  —a  x  ~2b. 

—3ax  X  —2km  x  —  7.r  x  —^hmx, 

—ny  X  (jy  x  —2y  x  3bm. 

xy  X  2y^  x  y^x  x  2ayx^. 

5  7/2  X  —  %?/  X  —2z^x  —ax^z, 

'utx  X  anx  X  '']z  x  bby. 

—  ibz  X  —xzx  —yz  X  agz. 
2chz  X  2xh  X  —z^  X  —b(jz\ 
—e^x  X  3a;  X  eb^  x  ay. 


44 


ALGEBRAIG  OPERA TIONS. 


I 


11 


1 6. 

17- 

1 8. 

19. 
20. 
21. 

22. 

23- 
24. 

25- 

26. 

27. 
28. 

29. 

30- 


—  2e  X  —2^  X  «  X  Z*:*;. 

—  4r)'.v;  X  'My  x  —2a^y  x  —  a:^. 
«\<;  X  —  ^//y^  X  ax^  x  —  a;'-^?/. 

rt,/;^  X   —?/^  X   —1  X  3«:c  X  — «*^. 

m\v  X  —  >^'^^  X  —inn^  x  w«.t  x  —  w^ 

— rti.«  X  —((jl'  X  «.?^"  X  a\c^. 

px'  X  (/'/  X  xi/  X  —  a.T. 

^^ic  X  —d^  X  «.i'^  X  —1  X  3«a:. 

ax  X  3c:6'  x  —  ,)W^r  x  —4i/^x  Gm. 


—  Guix  X  —2n^x  X  ;t«c  x 

u 


m^. 


— «  X  be  X  ~1  X  ^  X  3«2  X  4a:y  x  y. 

—1  X  ax  X  a^x  x  a^x^  x  bx  x  d. 
—an  X  2am^  x  —3nin  x  bn^y  x  —m. 
—mx  X  nx  X  —mil  x  — xy  x  — 1. 

—  2])x  X  —3qx  X   -m^x  X  jy^  X  — 1. 


Products  of  Polynomials  by  Monomials. 

73.  The  rule  for  multiplying  a  polynomial  is  given  by  the 
distributive  law  (§  G5). 

Rule.  Multiply  each  term  of  the  polynoinial  hy  the 
Dionornial,  (uul  take  the  algebraic  sum  of  the  products. 

Exercises.    Multiply 

1.  3.f^  —  4:xy  —  oy^  by  —  Aax. 

Ans.  —  12ax^  +  IGax^y  +  20«jyl 

2.  3.?:2  —  xy  4-  y^  by  3.1'. 

3.  x'  +  xy  +  7/2  by  3.t.  4.     ax  ■\-  by  -{-  cz  by  uxyz. 
5.     3fta.'3— 5r(?/2— 7  by  9r<5.r.      6.     4mp  —  Gnq  hy  —dmq. 
7.     oa^y^  —  ^a^y"^  —  7a'^y  by  Sab. 

74.  Tlie  products  of  aggregates  by  factors  are  formed 
in  the  same  way,  the  parentheses  being  removed,  and 
each  term  of  the  aggregate  multiplied  by  the  factor. 


MUL  TIP  Lie  A  TION. 


45 


,1 1 


dais. 

ven  by  the 

lal  hy  the 
roducts. 


3y  ((xyz. 
—  'dmq. 

e  formed 
ved,  and 
ictor. 


Example.    Clear  the  following  expression  of  parentheses : 

am  {a  —  b  +  c)  —  p  [a  —  {h  —  k)  —  ///  {<(  —  l))\. 

Bv  the  rule  of  §  73,  the  first  term  will  be  reduced  to 

ahn  —  amb  +  amc,  (1) 

The  aggregate  of  the  second  term  within  the  large  paren- 
theses will  be 

a  —  U  ■\-  k  —  m  (a  —  b) 

z^a  —  h  -\-  k  —  ma  +  mb,  {"l) 

because,  by  the  rule  of  signs  in  multiplication, 

—  m  [a  —  b)  =z  —ni  x  a  —  m  x  —b=z  —  ma  -\-  mb. 

Multiplying  the  sum  {'i)  by  —  p  and  adding  it  to  (1),  wo 
have  for  the  result  required: 

a^m  —  amb  +  amc  —  /ja  +2^^^  —  P^  +  pma  — ^;?«^. 

EXERCISES. 

Clear  the  following  expressions  of  parentheses  : 

1.  p  {a  +  m  —  p)  +  q(b  —  c)  —  r  {b  -\-  c). 

2.  {in  —  an)  x  -  -  {m  +  an)  y  -f-  {an  —  m)  z. 

3.  a  {x  —  y)c  —  b{x  —  i/)d-\-f{x-\-y)  cd. 

Here  note  that  the  coefficient  of  a;  —  y  in  the  first  term  is  ac. 

4.  am  [x  —  a{b  —  c)]  —  bn  \(ix  +  b{c  -{-  d)^^. 

5.  p\^— a{m  +  n)-\-b{m  —  n)^  —  q{b(vi  —  n)—a{m-\-n)\ 

6.  3.r  (2(7  —  nc)  +  ^  {^^^  —  3t')  —  z  {2m  -f  7n). 

7.  am  [m  {a  —  b)c  —  3h  {2k  —  id)  -f-  4;^]. 

8.  2pq  [3(7  —  ob  —  Gc  —pq  {2m  —  3;0]. 

9.  bn  [—  7«  -^(a  —  c)  —  (3  -a  —  b)\ 
10.  p{q  —  r)+q{r  —  p)-\-r{p  —  q). 

15.  The  reverse  operation,  of  summing  several  terms  into 
one  or  nu)re  aggregates,  each  multiplied  by  a  factor,  is  of  iVe- 
(pient  application.     Tiius,  in  >^  (!5,  having  given 

inp  +  mq  +  mr, 

we  express  the  sum  in  the  form 

m  {p  +  q  ■\-  r). 


46 


ALGEBRAIC   OPERATIONS. 


*i 


The  rule  for  the  operation  m 

If  the  sitni  of  seueral  terms  having  a  cmnninii  factor 
is  to  he  fomned,  the  eocfficiciits  of  this  factor  may  be 
added,  and  their  aggregate  he  niiiltiplied  by  the  faetor. 

Note.     This  operation  is,  in  principle,  identical  with  that  of  g  55. 

EXAMPLES. 

abx  —  hex  —  ady-{-^(lfj —  3dx-\-4ady  -f-  ?))>/ — ami/  —^cmx-\-bmx. 

Collecting  the  coefficients  of  x  and  y  as  directed,  we  have 
{ah  —  he  —  3/;  —  ^cm-\-bm)  x  -\-  ( — ad -{-^d  +  Aad  +  m—am)  y. 

Applying  the  same  rule  to  the  terms  within  the  parentheses, 

we  find 

ah  —  hc  —  U  =  h{a  —  c  —  3). 

—  ?icm  +  hm  =z  m  {h  —  36'). 

—  ad  +  od  +  Aad  —  dad  +  3d 

=  (Sa  +  3)  d 

=  d{a-{-l)d. 

m  —  am  =  m  (1  —  a). 

Suhstituting  these  expressions,   the    reduced    expression 
becomes 
[/>  (^  _  c  —  3)  +  in  (b  —  3c)]  X  +  [3  {a  +  1)  d  +  m  (1  —  a)]  y. 

The  student  should  now  be  able  to  reverse  the  process,  and 
reduce  this  last  expression  to  its  original  form  by  the  method 
of  §  ;-i. 

EXERCISES, 

In  the  following  exercises,  the  coefficients  of  y,  z,  and 
their  products  are  to  be  aggregated,  so  that  the  results  shall 
be  expressed  as  entire  functions  of  x,  y,  and  z,  as  in  §  55. 

1.  ax  4-  bx  —  3ax  +  3bx  +  G.r  —  7;?'. 

Ans.  {— %a  -\-  ^b  —  1)  X. 

2.  my  -\-  py  —  my  —  2py  —  3gy. 

3.  iiix  —  n.y  -f  2^x  —  gy  -\-  rx  —  sy. 

Ans.  {m  -i-  p  -^  r)  X  ~  (n  +  g  -}-  s)  y. 

4.  3az  —  y  —  ^az  -\-  z  —  az  -\-  y. 


lias 


I 


MUL  TIP  Lie  A  TION. 


47 


oil  factor 
r  VI ay  ho 
factor. 

t  of  §  55. 

•mx-\-bmx. 

we  have 
II— am)  y. 

rentheses, 


jxpression 

1  -  a)]  y. 

Qcess,  and 
e  method 


y,  z,  and 
suits  shall 

—  1)  X. 


i 


1 


5.  abxy  —  hcxy  -+-  bd.ry. 

6.  'Kxib.ry  —  'Z^x  —  ax  —  7xy. 

7-     ('!/  ~  %  —  ^^^^'1/  ~  '^^1/  +  '^'^' 

8.  rt/////  —  /J*/;///  +  any  —  buy. 

9.  ;>y2;  —  'iqrz  ~  A])j)z  +  Sf///z. 
10.     rw:K  -f  /'/^.v  —  (Dfiy  —  2bny. 

KJ.  An  entire  function  of  two  quantities  can  be  regarded 
iis  ;in  (Milirc  function  of  either  of  them  (§§  40,  50),  and  wlien 
expressed  as  a  function  of  one  may  be  transformed  into  a  func- 
tion of  the  other. 

Example.     The  expression 

{'>a  +  3)  x^  —  (4«2  _  2a)  .7.2  +  {n^  -  2a  +  \)  x  -  a^ 

luis  the  form  of  an  entire  function  of  x.     It  is  required  to 
ex})ress  it  as  an  entire  function  of  a. 
Clearing  of  i)arentiieses,  it  becomes 

2ax^  -\-  3.r3  —  4:a^x^  +  2ax^  +  a^x  —  2ax  -{-  x  —  a^. 

Now,  collecting  the  coeflticients  of  a^,  a^,  etc.,  separately,  it 
becomes 

(_  4.,;2  ^  x  —  1)  a^  +  {2x^  +  2.1-2  _  2x)  a  +  3^-3  +  x, 

which  is  the  required  form. 

EXERCISES. 

Express  the  following  as  entire  functions  of  y : 

I.       (3^2_4,^)^3_^(y_Oy2^1)^^.2  4.(2?/3  +  5?/2— 7).T— ?/2  — G. 
3.       (y5  _  2tf)  .x-3  +  (^4  _  2^2)  ^2  _|_   (^3  _  2?/)  X  -\.if  —  2. 

4.     {y'  +  3y^)  x^  +  (?/4  +  3 f/3)  X'  +  (^3  +  3y )  ,,2  +  (^2  _^  3 )  ^. 

3Iultipliccitioii  of  Polyjioiiiials  by  Polynoiiiiiils, 

17.  Let  us  consider  the  product 

{a  4-  b)  {p-{-q  +  r). 
This  is  of  the  same  form  as  equation  (1)  of  §  05,  (a  +  b) 
taking  the  place  of  7n.    Therefore  the  product  just  Avritten  is 
equal  to 

(a  -\-b)p  +  {a  +  b)q  -\-  {a-{-b)r. 


48 


ALGEBRAIC   OPERATIONS. 


K 


(       .V 


N    ^1. 


i.  ♦ 


«! 


But  {ii  4-  h)])  =  ap  -f-  Up. 

{a  -\-  b)q  =i  cq  -\-  hq. 
{a  -\-  d)  r  =  ar  +  br. 
Therefore  tlic  product  is 

aj)  +  hp  +  aq  -f-  Iq  +  ar  -f-  ir. 

It  would  have  been  still  shorter  to  lirst  cleiir  the  paren- 
theses t'roin  {a  -f  h),  putting  the  product  into  the  form 

« {P  +  '1  +  r)  +  h  {p  -\-  q  -\-  r). 

Clear! n<^  the  parentheses  again,  we  should  get  the  same 
result  as  ])efore. 

We  have  therefore  the  following  rule  for  multiplying  aggre- 
gates : 

1<S,  Rule.  Multiph/  each  term  of  the  inuUiplicdiKl 
hy  eacli  term  of  the  vndtijjJier,  and  add  the  products 
with  their  proper  algebraic  signs. 

EXERCISES. 

1.  {a  -f  h)  {2a  -  bn^  —  Vni^). 

2.  {a  —  b)  (3?/i  +  2m  —  r)abm7i). 

3.  {m^  —  9t^)  {2?nn  +  pm  +  qn). 

4.  {p"^  +  Q^  ^-  r^)  {P^l  +  (/'■  -t-  n^)' 

5.  (:1a  —  ^b)  {2a  f  2b). 

6.  {mx  —  vy)  {mx  +  ny). 

79.  It  is  frocpiently  necessary  to  multiply  polynominls 
containing  powers  of  the  same  letter.  In  this  case  the  begin- 
ner may  find  it  easier  to  anange  multiplicand,  multiplier,  and 
product  under  each  other,  as  in  arithmetical  multiplication. 

Ex.  I.    Multiply  7.r3 

The  first  line  under 
tlio  multiplier  contains 
the  products  of  the  sev- 
eral terms  of  tlie  multi])li- 
cand  by  "ii.v'^.  Tlie  second 
contains  the  products  by 
— 4.r,  and  the  third  by  —5. 
Like  terms  are  ]>]aced 
under  each  other  to  facil- 
itate the  addition. 


o. 


-  G.t'2  -f  5a:  —  4  by  32-2  —  ^x 

WORK. 

7.r3_G.^2  +  5a;— 4 
3.^2—4.?: —5 

— 28.6-4  ^  34;^-3_  2o./:2  .^  1  Cyj. 

—352:3  ^  30.r2— >ir).g  +  20 

21x5_46:c4^   4ar3_  2;<;2_  9^4.20 


^s 


MULTIPLICATION. 


49 


Ex.  2.    Multiply  m  +  nx  +  ;;2^  by  a  —  da:. 


r  tlic  pare II- 
furiii 


^t't  the  sumo 
plying  aggrc- 

ulfiplicaiul 
10  products 


polynomials 
B  the  begin- 
iltiplier,  and 
iplicatioii. 

-  ^x  —  5. 


}-\C)x 

-   9.T  +  20 


m  4-  7/.^  +  2>x^ 
a  —  bx 

am  4-  a}ix  -f-  «;o.f2 
—  bmx  —  bnx^ 


bpx^ 


am  4-  {a7i  —  bm)  x  +  {ajJ  —  Z'/i)  ^  —  //jya--^ 

In  the  following  exercises  arrange  the  terms  according  to 
;iie  jiowers  and  products  of  the  leading  letters,  a,  b,  x,  y,  or  z, 

MuUiply 

?ui^  +  5rt  +  7  l)y  'id-  —  'ia  +  4. 


I 

2 

3 

4 

5 
6 

7 
8 

9 

10 

II 

12 

13 

14 

15 
i6 

17 

i8 

19 

20 
21 
22 

23 
24 

25 
26 


^/^  +  r;6  4-  b^  Ijy  «  —  J- 
(fi  4-  rf^  4-  r/./'-^  4-  .^-3  by  r^ 


.T. 


a' 


a' 


4-  r/  —  1  by  a^  —  a  -\-  1. 


.6-*  +  ax^  4-  «''.?:2  -|-  (i?x  4-  «.■*  by  :r  —  a. 

a  4-  Z'^  4-  cz^  4-  ^/;2;^  by  m  —  nz  4-  ^2;^. 

3^2  f  5^  4-  7  by  2r;2  +  3«  —  4. 

«'-  —  ab  +  b^  by  «  4-  Z*. 

a^  4-  r/2^  4-  r<a;2  ^  x^  by  a  —  x. 

a'^  —  a'^  -\-  a  —  1  by  <«2  ^  ^  _  1. 

ar*  4-  ax^  4-  alT^  _|_  ^3^.  ^  ^4  ijy  a;  _j_  «. 

r^^  4-  J2;  4-  cz^  4-  ^/;2;-^  by  wi  4-  nz  —  /^^^^ 

(rt  4-  J.?;)  (?w  4-  ?<a:). 

[a  4-  Z*.?:  4-  c.f2)  {m  4-  wa;  4-  j?:^^). 

[if  -  ^  +  2)  if  -  2). 

(//^  4-  2/'  4-  2/  4-  1)  (i/2  4-  Z/  4-  1) 

{y'  -  '^f  +  3//  -  4)  (/  +  2/  4-  3^  4-  4). 

3rt>a;  —  3«2^  4-  'id^"  by  «"'  —  a" 

a^  4-  Grti  4-  ., b  by  r«  —  -b. 
3       -^  3 

(f/  +  /;)  4-  {((  —  b)  by  {a  +  b)  —  {a  —  b). 

(,2  _  Z,3  +  (rt  _  ^,)    by  f^2  _^  ^,2  ^  (,^  ^  ly 

a  -\-  b  -\-  c  hy  a  —  b  -\-  c. 

«2  ^  J3  _  (3^2  ^  J2)   by  2«  +  2/^  —  2  («  -  Z*). 

'Z  (a  —  b)  -]-  x  —  y  hy  a  -\-  b  —  {x  +  y). 

ax""'  4-  Z»^"  —  abx  by  «;/;2  4-  Z/a:3^ 

«"'  —  d>"  by  «"'  4-  b". 


50 


A LGEBRAIC    OPERA  TIONS. 


A 


27.  _  ir,.A/  +  Wxif  -  I'Mf  by  -  hxy, 

28.  JJ^-2  .^  3^,.,.  _  ^^2  i)y  o.r'  —  ^/.r  —  -a2. 

NoTK.  Aggregates  cnteriijg  into  either  factor  should  bo 
sinipliliocl  bel'ore  niultii)l}'iiig. 

Si)OciJil  Forms  of  Multiplieatioii. 

8().  1.  To  find  the  square  ol'  a  biuoniial,  as  a  +  h.  We 
multiply  a  -\-  b  hy  a  -\-  b. 

a  (a  -f  />)  =  a^'  -]-  ah 

h  (a  +  b)  —  ab  +  b^ 

(a  +  b)  {a  4-  b)  =  ifi  +  'Zab  +  b^ 
Hence,  {n  +  bf  —  a^  +  'lab  -\-  b^  (1) 

2.  We  find,  in  the  same  way, 

{a  -  bf  =  a^  -  2ab  +  bK  (2) 

These  forms  may  be  expressed  in  words  thus: 

Theorem.  The  square  of  a  binomial  is  equal  to  the 
sum  of  the  squares  of  its  two  terms,  plus  or  minus  twice 
their  product. 

3.  To  find  the  product  of  «  +  ^  by  a  —  b. 

a{a  -\-  b)  =  a^  +  ab 
—  b(a  +  b)=      -ah-W 
Adding,    {a  +  b)  {a  —  b)  =  'cfi  —  b\  (3) 

That  is; 

Tlxorem.  The  product  of  the  sum  and  difference  of 
two  numbers  is  equal  to  tlie  difference  of  their  squares. 

The  forms  (1),  (2),  and  {'**>)  should  be  memorized  by  the  student,  owing 
to  their  constant  occurren';e. 

When  i  =  1,  the  form  (3)  becomes 

{a  +  1)  («  -  1)  =  «2  _  1, 

The  student  slio:ild  test  those  formulae  by  examples  like 
the  following: 

(9  +  4r^  =  92  4-  3.9.4  +  42  =  81  +  73  +  16  =  169. 
(9  -  4)2  =  92  -  2.9.4  +  42  rz:  81  -  73  +  16  =.  35. 


41 


r 

ineni 

I 

3 

5 
7 

S 

it  l\)l 

li 

11 

■/ 

a  —  ' 

S 

1] 
1 


nc^fi 


whic 

this  I 

wliie 


1 


MULTIPIJCATIoy. 


51 


should  b(j 


1. 


+  b.     We 


(1) 

ml  to  tlio 
iius  twice 


(3) 

3rence  of 
squares. 

dent,  owing 


nples  ijke 

169, 
25. 


I 


(f,  +  4)  (0  -  4)  =  1)2  -  4-  =  C5. 

Prove  these   three  e(iiuition.s  by  computing  the  left-luind 
nieuil)er  cUrecLly. 

EXERCISES. 

Write  oil  sight  the  viiUies  of 

I.     {lit  -f  -^n)^.  2.     (m  —  2nY. 

5.  C^^:  +  ij)  {2x  -  y).  6.     {'.Ix  +  1 )  {:]x  ~  1). 
7.     (4.t2  +  1)  (4a;'^'  -  1).  8.     (5rJ  -  3)  (o.f^  -f  IJ). 

SI.  necaupo  the  product  of  two  negative  factors  is  positive, 
it  follows  lluit  the  s((u;ire  ot*a  negative  (puuitity  is  positive. 

Examples.      (—  of  =  a-^  z=  {-\-  af. 

{b  —  r/)2  =  (fi  —  2ab  +  Z>2  =  («  -  by. 

Hence, 

'f/ie   (!xprcc?ir„i  a^  —  '^ab -\- Ir  is   Ihc   Sijiuivo  both   of 
((  —  b  (Hid  of  b  —  a. 

S*i.  We  have  —  a  x  a  ^=  ■—  a^. 

Hence, 

TJie  product  of  equal  factors  with  opposite  signs  is  a 
nc'J>(ibive  square. 

Example.       —  {a  —  b)  {a  —  b)  =  —  (fi  -f  2r/J  —  Z^l 

whicli  is  the  neg."^ive  of  ('2).     Bet, use  —  {a  —  b)  =z  b  —  a, 
this  efpiation  may  b^  written  in  the  form, 

(J)  _  a)  {a  —  b):=—n^-{-  2ab  -  b\ 
which  is  readily  obtained  by  direct  multi[)lication. 

EXERCISES. 

Write  on  sight  the  values  of 

1.  —{a  +  b)  X  —  (rt  +  b). 

2.  {x  -  y)  (//  -  x).  3.     (.r  +  y)  (-  X  -  //). 

4.     (2rt  —  Zb)  (3^  —  2rt).  5.     {U  —  2a)  (—  '6b  +  2a). 

6.  (am  —  hn)  [bn  —  am).         7.     {xy  —  2)  (2  —  xy). 


1>» 


G2 


ALOKniL  1 IC   OPERA TIONS. 


CHAPTER    ill. 


DIVISION. 

83.  Tlio  prohlom  of  alf^obniic  division  is  to  find  huch  an 
oxpivssion  that,  M'lu'ii  multiplied  by  llio  divisor,  tlie  product 
Hliiill  be  the  dividend. 

This  oxi)ression  is  called  the  ([uotient. 

]n  Ali(ebni,  the  (jnotiont  of  two  quantities  may  always  be 
indicated  by  a  fraction,  of  which  the  numerator  is  the  divi- 
dend and  the  denominator  tho  divisor. 

Sometimes  the  numerator  cannot  be  exactly  divided  by  the 
denominator.  The  expression  must  then  be  treated  as  a  frac- 
tion, by  methods  to  be  exjdained  in  the  next  chapter. 

Sometimes  the  divisor  will  exactly  divide  the  dividend. 
Such  cases  form  the  subj(>ct  of  the  present  chapter. 


# 


'I, 


} 


Division  of  Moiioniiiils  by  3roiioniijil.s. 

81.  Ill  order  that  a  dividend  may  bc^  exactly  divisi- 
ble by  a  divisor,  it  is  necessary  that  it  shall  contain  tlie 
divisor  as  a  factor. 

Ex.  I.     15  is  exactly  divisible  by  I),  because  3'5  i=  15. 
2.  The  product  ab'c  is  exactly  divisible  by  ac,  because  ac  is 
a  factor  of  it. 

To  divide  one  expression  by  another  which  is  an  exact 
divisor  of  it: 

IvULE.  BcmovG  from  the  (Jividencl  those  factors  the. 
])ro(hict  of  ivhicli  is  equal  to  the  divisor.  The  reiiiaiu- 
zn£>  factors  will  he  the  quotient. 

8.1.  Ride  of  Exjwnenls.  If  Loth  dividend  and  divisor 
contain  rbe  same  symbol,  with  different  exponents,  say  m  and 
n,  then,  because  the  dividend  contains  this  symbol  m  times  as 
a  ftictor,  and  the  divisor  n  times,  tho  quotient  will  contain  it 
m  —  n  times.    Hence, 


'It^ 


1)1  VISION. 


m 


lul  .such  an 

lie  i)ro{luci 


y  always  be 
is  the  (livi- 

idcd  by  the 
d  as  a  f  rac- 
er. 
e  dividend. 


5tly  divisi- 
oiitaiu  tlie 


)  =  15. 
ccausc  ac  is 

is  an  exact 

'^actors  the 
e  rcDiaiii- 

md   divisor 

,  say  m  anil 

m  times  as 

contain  it 


In  iVividin^,  c.vpnticntfi  of  like  syDihoU  are  to  he  sub' 

EXERCISES. 

1.  Divide  'l^lnj  by  'iy.  Ants.  lo.c. 

2.  Divide  '^U(^c  by  Ik. 

3.  Divide  j^  by  x^.  Ans.  x. 

4.  Divide  \M'  by  Grt.  Auk.  Wa. 
'$        5.  Divide  Uui'in  by  3rt.  Ans.  bam. 

6.   Divide  Ib/W  l)y  '^nm. 
I        7.   Divide  l(VA/*4  by  8rrW.        8.  Divide  ^iO.rifz^  by  O./v/^. 
9.   Divide  •lOa'-V'^s  i)y  10aV2«.    10.   Divide  '^oul/^  hy^7nL 

Rule  of  Sij?i!s  ill  Division. 

8(>.  Tbe  rule  of  .signs  in  division  corresponds  to  that  in 
multiplication,  namely: 

Ifdiridciul  ftnd  divisor  have  the  same  si^ii,  the  (fuo- 
iiciit  is  positive. 
I        //  tJicy  have  opposite  signs,  the  qaotienb  is  negative. 

Proof. 

-\-mx  -^  (4-^/')  =  +:i»',  because  -\-x  x  {-\-'>n)  =  -[-nix. 

'.         -\-hix  -^  {  —  hi)  =  —X,       "  —X  X  {  —  )'i)  =  i-)iix. 

#        —mx  -r-  (  +  iii)  =  —^',       "  —X  X  (  +  »')  =  —nix. 

—  mx  -^  {-ni)  =  -[-X,        "  -\-x  X  (  —  hi)  =  —in.r. 

The  condition  to  be  fulfilled  in  all  four  of  these  cases  is 
tluit  the  product,  qiioiicfit  x  dicisor,  shall  have  the  same  alge- 
braic sign  as  the  dividend. 


EXERCISES. 


Divide 

I. 

+ 

a  by 

+  «. 

2. 

+ 

a  by 

—  fl'. 

3- 

— 

a  by 

+  n. 

4. 

— 

a  by 

—  rt. 

5- 

— 

'.y.^ifiuix  by 

ilax. 

6. 

24:X^l/z  by 

12xyz. 

7- 

2b 

tin^x'" 

H  - 

-  lamx^. 

Ans.  4  1. 

Ans.  —  1. 

A)ix.  —  1. 

Ans.   +  1. 

Ans.  —  Mm. 

Ans.  —  2x. 

Ans.  —  3?nx"'-\ 


54 


ALGEIillAW   OP  Eli  A  TIONS. 


8. 
9- 

lO. 

II. 

12. 

13- 
14. 

'5- 
16. 

17. 


—  18rt"*/;«  by  —  (;«"y;.  Ans.  3«»^-«;;»-i. 

—  UkfixP^yn  by  Aax'ij'K 
Ul/pi  h\  —  To'' pi. 

—  VZb>'H"k-"'  by  —  Ah^l^kn, 

12  (a  -  //)'^  r*  by  13  (-  -  /y)2  r.        Ans.  4  (r^  -  ^.)  c^ 
42  (a:  —  //)»'  by  —  7  (.7;  —  yy. 
_  44rt''  (7:  -  iiY  ])y  11^/'  (.f  -  yy. 

—  48  {m  +  7)y>  ])y  —  8  (m  +  71)'^. 

04  (a  +  hy  {x  -  yy>  l)y  4  {a  +  /y)  (.?:  —  y). 


It    I 


■='»     ! 


Division  of  I*i,lyiioiiiiiils  by  Monomials. 

87,  By  the  distriljutive  law  in  miilti[)licati()ii,  whatcvci' 
quaiitiiics  tiie  symbols  in,  a,  b,  c,  etc.,  may  rei)resenl,  we  have: 

{((-]-  h  -¥  c  -\-  etc.)  X  m  =  ma  -\-  vih  +  mc  +  etc. 
ThiTcfore,  by  the  condition  of  division, 

{ma  +  mh  +  mr  4-  etc.)  -^  m  =:  a  ^  b  -\-  c  +  etc. 
Wc  therefore  conclude, 

1.  In  order  that  a  polynomial  may  be  exactl}'  divisi- 
ble by  a  monomial,  each  of  its  terms  must  be  so 
divisible. 

2.  The  quotient  will  be  tlie  algebraic  sum  of  the 
se])arate  quotients  found  by  dividing  the  different  terms 
01  the  polynomial. 

EXERCISES. 

Divide 

I.     :>r/2  +  (vAr  —  M^x^  by  2^/2.      j{j^^^  i  ^  3^^^.  _  ^^3^.3^ 

i\ni'^u  —  Vhn^n^  —  \'6mn^  by  ijmn. 
<\,(%^  _  i(\a'^i,i  4.  .Sr/s^-'i  by  4^/3^3, 

4r//''  —  ^x^y^  4-  Ax^ii  by  —  4.r//. 

Viabx  —  U(tbx'^  by   —  \2abx. 

^lamh""  —  Ua^uiir^'  +  2Sa^ni''x^  by  —  ^atnx^ 

'\'ia'ir  4-  2Aax  +  48r?.?-2  by  24r/r. 

a  (b  —  (•)  -^  b(c  -'  a)  -{-  r{a  —  b)  +  abe  by  abc. 

27  (^^  _  by  -  18  (rr.  -  />)'  +  9  {a  -  by  by  "o  (a  -  Z»). 

r/»*  (rt  —  Z*)"  —  r««  («  —  b)"^  by  ^i"  («  —  by. 


2. 

3- 

4- 

5- 
6. 

7- 
8. 

9- 
10. 


i 


^    N'V, 


Dl  VISION. 


5i5 


m-n  ))7i-l 


f 


-  h)  c\ 


ials. 

whatever 
.,  we  liave : 

-  etc. 

-  etc. 

tlj'  divisi- 

st  be   so 

n  of  i\w 
ent  terms 


—  4:a^xK 


abc. 
{a  -  h). 


J 


1 1 . 

12. 


I.)- 


(,/  4_  IjY  {a  -  by  +  {a  +  by  {a-hy  l.y  {a  +  b)  {a-b), 

10  (i;  +  yy''{x  -  y)'^  -  h  (./•  +  y)''  (-^  -  nf 

by  5(.i'  +  i/)  (-'•-//). 
(,,  +  /,)  (,r  _  /,)  l,y  (fi  —  b\ 


Factors  and  3Iultii)les. 

88.  As  in  Aritiimetic  some  iiiun])ers  arc  composite  and 
otliers  prime,  so  in  Alg('l)ra  some  expressions  admit  of  beinu^ 
divided  into  algebraic  factors,  wliile  otiiers  do  not.  The  latter 
are  by  analogy  called  Prime  and  the  former  Composite. 

A  single  symbol,  as  a  or  x,  is  necessarily  prime. 

A  product  of  several  symbols  is  of  conrse  composite,  and 
can  be  divided  into  factors  at  sight. 

A  binomial  or  polynomial  is  sometimes  ])rime  and  some-, 
times  composite,  but  no  universal  rule  can  br  given  for  dis- 
tinguishing the  two  cases. 

8i).  When  the  same  symbol  or  expression  is  a  factor  of  all 
the  terms  of  a  polynomial,  the  latter  is  divisible  by  it. 


I. 

2. 

J- 


EXAMPLES. 

ax  -\-  (ibx^  4-  a\'x^  =  a  {x  +  bx"^  +  (fc^). 
((^lAc  +  (t^U^x^  =  (tVA<:  (b  +  ax). 
«"^"  +  (("X"  =  a't  {a'i  +  X"). 

EXERCISES. 

Factor 

I.     (ix^  +  ((^x.  2.     aWcy  -\-  aVjc^j/  -}-  abh^ij. 

3.     r<''«  b''  4-  ««  b-''.  4.     «3"  .f"  —  «2n  .r""  +  ««  .^•3«. 

5.     a''  b'"  c^'^  +  a^'^  b^"  c"  +  rt^w  fjn  c'in, 

*,}(}.  There  are  certain  forms  of  composite  expres.'-ions 
which  should  be  memorized,  so  as  to  be  easily  recognized. 
Tile  following  are  the  inverse  of  those  derived  in  §  80. 

1.  (/2  +  '^ab  +  b'^  =  {a  +  b)\ 

2.  a'i  —  •>ab  +  b^  =  {a  —  b)\ 

3.  r/2  -  />2  ^  (,,  _^  /,)  (,,  _  ^,). 

I'lie  form  (3)  can  be  api)lied  to  any  difference  of  even 
powers  ;  thus, 


s' 


56 


ALGEBRAIC   OPERATIONS. 


I* 


"1 


wi 


a' 


«' 


and,  ill  otiictuI, 


a- 


-b*  =  (^2  +  P)  («2  _  /,2)  . 

—  /j'^  =  {a-^  +  0^)  {a-^  —  ^3)  . 

—  0'-"  =  {(>"  -\-  6")  {(("  —  />"). 


If  the  expoiic'iit  icj  u  iiuiUij)l(.'  oi'  4,  llic  second  luc-tor  can  be 


airain  divided. 


EXi^.  MPLES. 


a^  -1/  :=  {a^  +  b^)  {a^  -  t^)  =  {d'-^b'')  {a  +  b)  (a-b). 
a^  -b^=  [a^  +  b')  {a^  -  b')  ^  {a^-^-b^)  (a'^-^b^)  {a-}-b)  (a-b). 
When  b  is  equal  to  1  or  2,  the  forms  become 

a^-1  =  {a  +  l)(a-l). 

d^-4:  =  [a  +  2)  (a  -  2). 

(i^  4-  2rt  +  1  :=  [a  +  ly. 

d^  -t-  Aa  -\-  4:  =z  {a  +  2)-'. 

ai  _  o^^  ^  1  ^  (^^  _  1)2  ^  (1  _  a)\ 

a'  —  4^/  +  4  =  {a  —  2)^  =z  (2  —  ^^)2. 

By  initling  2b  for  Z*,  they  f'ive 

a'  -  4/y2  z:z  {a  +  2b)  (a  -  2b). 

a2  _j.  4ab  +  4/^2  zr:  («  -|-  2^)2. 


EXERCISES. 

Divide  the  following  exi 
po.ssible  : 

)ressi()ns  into  as  many  factors  as 

I. 

2. 

^•4-10. 
yi  —  Ux*. 

A71S.  (x'2-f-  4)(.i-  +  2)(:i;-2). 

3. 

4. 

x^  +  ()X  +  0. 

2-2          i]z  -]-   0. 

^«i\    (.T  +  3)2. 

5- 

4rt2.c2  _  dby. 

6.     1G^^1j-4  _  i. 

7- 

9- 
II. 

13- 

9.?;2  _  I2rf/  +  4^2. 
4^2.^2  +  4ub.i-y  +  %l 
r/-4  —  2.c'y  4-  7/4. 

8.     «2<.2  ^_  2^.ry  4-  f, 

lO.       rt»  +  4^/2^-2  ^  4^2, 
12.       .9-4  —  4.t'2^2  _|_  4^4^ 

14.     «*      a%\ 

IS- 
1 8. 

a2n  _  o^^ft  _|_  1^ 

1  -  ;y4. 

3^Z  +  2;?:3y3^  _^  ^^y6^^ 

16.      .t2"  —  4rt.l'«  +  4c/2. 

J7ifi.  z{: 

^6  +  2Ty  -{.  y^)    =  Z  (j-3  4-  2/3)2. 

DIVISION. 


57 


•r  ciiii  be 


){a-b). 


I 

•if 


2-.     a-iw  _  'Zx-i'^yn  ^  ^-z*.  26.     aH'«  —  ;>./;•-''"  +  1. 


27.     .f2  +  a;  +  ^-  28.     x'2'«  +  ./;'"  4-     • 

*.)1.  By  combining  tlie  preceding  forms,  yet  other  forms 
iiiuy  l)e  found. 

For  example,  the  factors 

(di  4.  ah  +  Z-^)  (^2  -  ab  +  Z/2),  (i) 

are  respectively  the  sum  and  diU'erence  of  tlie  quantities 

«'  +  li^     and     ab. 

Hence  the  product  (I)  is  equal  to  the  difference  of   the 
f^quares  of  these  quantities,  or  to 

Hence  the  latter  quantity  can  be  factored  as  follows: 


Lctors  as 

-2). 

+  3)2. 


f  y^)\ 


■& 
■^ 
s 


Factor 

I.     x^  +  xhf  4-  ?/4. 


EXERCISES, 


2.       ^4+  8rt2Z>2  4.  ir,/A 

4.     rt^"  +  d''^  U^"  +  Z/'". 

6.       ««  +  8rt4^2  ^  1(;^^2J4. 


3.     r(4  +  9^<2i'2  +  sir". 

5.       rtU-2  4.  4^;2^2,.2  ^  1(3^4^2. 

7.     r<;5«  +  x^^  f^  +  2;^'  //'«. 

8.       ?//2  __  ^2  ^_  O^//;  _  /^2_         J^;^s.^     (;;^  —  rt  +  Z/)  (//<   +   r/  —  //). 

Here  the  last  tliree  terms  are  a  negative  s(iuare.     Compare  j;  82. 


9.       «2  _  4^2  _[_  4^^.  _  ^.2,  10^       ^,3  _  4f,^2  _j_  4^,^^.  _  ^/,.2_ 


1)'^.  The  following  expression  occurs  in  investigating  the 
;u'e;i  of  a  triangle  of  which  the  sides  are  given  : 

{a-\-b^  c)  {a^b-  c)  (.'«  -b-\-  r)  {it  -  b  -  c),        (1) 

By  §  80,  3,  the  product  of  the  first  pair  of  factors  is 
{(I  +  hf  _  ^.2  r=  ^2  4.  'l(^JJ  -t-  ^  -  f2 ; 

and  that  of  the  second  pair, 

{a  -  bf  -  r2  =  r?2  _  "lab  +  i'^  _  r^. 


58 


A  L  a  EBRA  W   OPERA  TIONS. 


By  the  smiie  principle,  lliu  product  of  tlicsu  products  is 
(«2  +  Z,2  _  ^..)3  _  4.^2^2^ 

uliicli  we  readily  find  to  be 

a^  ^  (jx  +  ,.1  _  -Zdib"'  —  2/A'^  -  Jir2(i2.  (2) 

Hence  this  expression  (2)   can   be  divided  into  the  four 
factors  (1). 

Factors  of  Binomials. 

(>3.  Let  us  multiply 

OI'EUATION. 

a;  —  a 


rt« 


Prod.,  x"^        0 


0 


0 


0 


«" 


The  intermediate  terms  all  cancel  eacli  other  in  the  product, 
leaving  only  the  two  extreme  terms. 

The  p'^'oduct  of  the  muUi])lieand  by  x  —  a  is  therefore 
^  —  iV^.  Hence,  if  we  divide  a;'*  —  i(^  by  x  —  a,  the  (luotient 
will  be  the  above  expression.  Hence  the  binomial  x^^  —  «" 
may  be  factored  as  follows : 

-  ri"  =  {x  —  a)  (:^;«-l  +  r^r"-2  4.^/^,•«-3  4_ ^a^-z^^a'^-i). 


X 


TV 


riierefore  we  have. 


Theorem.  The  dift'erence  of  any  power  of  two  niiin- 
bers  is  divisible  by  tlie  difference  of  the  numbers 
themselves. 

Tllustuatiox.  The  dilYerence  between  any  power  of  T 
and   the   same   power   of  2   is   divisible   by  7  —  2  :=  5.     For 

instance, 

72  _  22  =:      45  =  5.9. 

73  _  23  -    335  =  5.G7. 
7'«  -  2^  =  2385  =  5.477. 

etc.  etc.         etc 


04. 

x"~ 

Ny    X  -i 

Ki:.M 

ilint    — 

,  icllieiei 

:   n.)\vers  » 


;         The 

:;   arcurdii 


X 
X 


I'rod.,  X 

I    III   uiult 

%   onus   — 

lh>n« 

\   ;ulniits  i 

i   .'"'  -  (- 
i        If  a 

[in 


V*' 


1*  ^i'he 

I  Th(: 

tit 

i  uivisih 


niVL^lON. 


m 


cts  is 


the  four 


^  x  —  a. 

—  a» 

—  rt^ 

l)roduct, 

therefore 
quotient 
X"'  —  «« 


^o  nuni- 
umbei's 

ver  of  i 
:  5.    For 


<)4.  Let  us  multiply 

,v    .1-  +  rt  =  X  —  {—  a). 

I  Ki:.M.  1Miis  expre-siou  is  exactly  like  the  i^'ecediug,  rxeept 
f  il,;,(  ._  ^  IS  substituted  for  ^.  It  will  he  uotieed  tiuit  liie 
I  (  iclKeients  of  the  powers  of  x  in  the  niultiplieund  are  I  he 
j),)\vers  of  —  a,  because 

{-af  =  +ai, 
( -  ^0'  -  -  ^^^ 

etc.  etc. 

The  sign  of  the  last  term  will  be  positive  or  negative, 
iururding  as  n  —  1  is  an  even  or  odd  nun^ber. 

OPERATION. 

:/;'*  -1 — rta;«-"2  _j_  dic'^-^—a^c'^-i  4. ....  +  ( _  ^^)«  -2  ;r  +  ( _  a)''-^ 
a:  -j-  a  =  X  —  ( —  a) 

+  «a;^~^  —  ^'^^-""^  +  a^x^-'^ ....  —  (— (/)^^-^  .6-  —  {—(!)" 


0 


0 


0 


0 


k 


I 

■4 


I'rud.,  j« 

The  multiplier  a;  +  a  is  the  same  as  x  —  {— a)  (§  ■)'.)). 
Ill  nmltiplying  the  first  terms,  we  use  +  ((,  and  in  the  last 
onus  —  (—  ((),  because  the  latter  shows  the  form  better. 

Hence,  reasoning  as  in  (1),  the  expression  x"- —  {— (/)"' 
adniits  of  being  factored  thus  : 

/"  —  (—  aY'-  ^  {x  +  a)  [.6"  -1  —  (ix>^-^  +  ah'>-'^  — 

....  4-  (— rt)"'^.''  +  {—(')"~^]- 
If  n  is  an  even  number,  then  ( —  (z)'^  =  (t'^,  and 

a;«  _  (_r/)"  =  ./;«  —  (I'K 
if  N  is  an  odd  number,  then  (—  a")  =:  —  rt«,  and 

a?»  —  (—  (()"'  —  xJ^  4-  ri". 
Therefore, 

Theorem.  1.  When  vi  is  odd,  the  binomial  x^^-\-a^  is 
divisible  by  x-^a. 


60 


A  LGEBRAIC   OP  ERA  TI0N8 


^ 


I  I 


i 


^    ' 


TJieorem  2.  When  n  is  even,  the  binomial  x'^—w^  is 
divisible  by  x-\-a. 

Note.  T'.iese  theorems  could  have  been  deiUicecl  imme- 
diately from  that  of  §  03,  by  changing  a  into  — (i,  because 
X  —  a  would  then  have  '»een  changed  to  x  -\-  a,  and  x^  —  ^/'' 
to  X"'  4-  11"-  or  2"  —  a'\  according  as  )i  was  odd  or  even. 

The  forms  of  thu  laetors  in  the  two  cases  are  : 

AVhen  n  is  odd, 
x"^  +  aP  —  {x  +  a)  (a;»-i  —  rt.r^-^  ^  ^-x^-z  _  . . . .  -frtn-i). 

When  n  is  even, 

xn  _  ffU  z=  {x  ^  a)  (.t;"-l  —  fir'^-'i  4-  «2i.n-3  _  . . . .  ._^^n-l).       (^,) 

In  the  latter  case,  the  last  factor  can  still  be  di\  idcd,  Ijc- 
canse  x^  —  a'^  is  divisible  by  x  —  a  as  well  as  by  x  +  a.     We 
find,  by  multiplication, 
{x  —  a)  {x"-'^  +  rt22-w-4  _j-  a^x^-^  +  ....  +  fC^-^) 

—  x""-^  —  ax^-^  +  «2.^-«-3  —  a^x^-^  +  ....  +  a"-lv  —  a«-i. 

Therefore,  from  the  last  equation  (a)  we  have : 
When  n  is  even, 
x^  -  rt«  =  {x-\-a)  (x—a)  (^"-2-|-rt2.c«-4+  a^x^-<^  —  ....  +  a""-'). 

EXERCISES. 

Factor  the  following  expressions,  and  when  they  are  purely 
numerical,  prove  the  results. 


I. 

52  -  ^l 

Alls.  (5  +  2)  (5      2). 

[P^ 

'OOf. 

r.2  02 

rr 

■  4  =  21  ; 

(5 

4-  2)  (5  - 

2) 

!»,' 

.3  =  21.] 

2. 

53       23. 

3. 

5'       24. 

4- 

55  —  25. 

5. 

5<5  -  26. 

6. 

73     1     03 

7. 

73  _  23. 

8. 

i      •   -  /V  . 

9- 

X^  —  rtl 

lO. 

a;3  —  a^. 

II. 

:i;4      «4. 

12. 

x^  —  a^. 

13. 

.7'3  -f-  (fi. 

14. 

x^  +  ci\ 

15. 

a^  —  8^/3. 

16, 

8rt.3      27J3. 

17. 

IGrt'  -  ¥. 

18. 

:l-3  +  8/. 

19. 

:ri  —  1(1^4. 

20. 

8r/3  +  rih\ 

21. 

.7-fi  _  <i4r/«„ 

;    1)5. . 

tit's  is  a 
jiu'tors. 

i-iXAM 


iiiu'  ijiuinl 


|llii'  expre 
let'/,  nor 
if  actor. 


Ihf. 
y([ujuititi« 
'(h'gi'ee. 

KrLE 

(jiiiiiititi 

if  th 

Icoininoib 

I       J/'  Wi 

('(I  el  I  of 

%li((H  ill  (I 

I 

Ex.  I. 


The  f 
;:/'■',  while 

hx.  2 

Facto 

^  (^  +  //)  (, 
:       By  tl) 


DIVISION. 


61 


jed  imnie- 
/,  because 
(I  .6"  —  a'' 


ividetl,  br- 


..  +  «"--). 

are  purclv 
(5  -  2). 


Least  (U>iHnioii  3Iultii)lc. 

:■  <).">.  Def,  A  Cornmon  Multiple  of  several  quanti- 
itirs  is  any  expression  of  which  all  the  quantities  aiv 

liiu'toi'S. 

$  iix.vMPLE.  The  expression  arti^n^  is  a  c(fmnion  innhiple  of 
liiir  quantities  «,  m,  n,  am,  amn,  ain^,  m'n%  etc.,  and  finally  of 
lilir  expression  itself,  ani^nK     But  it  is  not  a  multiple  of  r/^,  nor 

'i<>t'/.  uur  of  any  other  s'-ubol  which  does  not  enter  into  it  as  a 

i 
llactor. 

ihf.    Tlie    Least    Common    Multiple   of   several 
([unntities  is  the  common  multiple  wliich  is  of  lowest 


+  a.    Wu      .^dcLcree.     It  is  written  for  sliortness  L.  C.  M. 


I      KuLE  FOR  FINDING  THE  L.  C.  M.      Fitctoi'  the  seuevul 
lijiiiiiiHtics  as  far  as  possible. 

'      7/  tJie  quantities  have  no  common  faetor,  the  least 
%c()innion  niultiple  is  their  jjrodact. 

5      //'  several  of  the  quantities  lave  a  coimnoib  faetor, 
Hltr  uinltiple  required  is  the  jn'oduet  of  all  the  frtetors, 

cncJi  of  til  em  being  raised  to  the  highest  power  which  it 

Jtds  in  any  of  the  given  quantities. 

Ex.  I.     Let  the  given  quantities  be 

^ab,        We,        (Sac. 

'4      The  factors  are  2,  3,  a,  b,  and  c.    The  highest  power  of  h  is 
\lf',  while  a  and  c  only  enter  to  the  tirst  power.     Hence, 

I  L.  C.  M„  =  Qal)^c. 


% 


Ex.  2.     «2  _  j2^  ^2  ^  2ab  -f  b\  a^  —  2ab  -f  b^  a^  —  b*. 
I'aetoring,  we  find  the  expressions  to  be, 
I  {a  -^h)(a-  b),     {a  +  b)\     {a  -  bf,    {a^  +  ^M«  +  h)  {a  -  h), 
V>)-  the  rule,  the  L.  C.  M.  rerpiired  is 

{a^by{a-bf{a'^-\-y^). 


62 


ALUhUIiAW   OPJiJIiA  TIONS. 


EXERCISES, 

Find  the  L.  C.  M.  of 


2.     d^b,  V^c,  chl,  d^cu 
4.     d^,  ab^,  bc'K 


I.  i)'i/,  :rz,   ijz. 

3.  a,  (lb,  ubc,  abed. 

5.  a.-2  —  if,  X  +  t/,  x  —  y. 

6.  x"  —  4,  -c'  —  Ax  +  4,  .^2  -f-  4a:  4-  4. 

7.  1(1^(1^2  —  4^/1^^  "^ax  -\-  in,  2ax  —  ?ii. 

8.  r^  —  1,  .T^  +  1,  .r2  —  :.^A'  +  1,  x"-  +  2x  +  1. 

9.  4r/  {b  +  r),  Z*  (r^  —  r),   '^r/5. 

10.  :3  {a     ■  by,  V  («  f  Z;)2,  ^  (r«  —  b)  {a  +  Z*). 

11.  3{x-      •»       (a:-//),  3(^;3  +  y3). 

12.  a  —  b,    "^   -  '      «^  —  Z*^,  rt*  —  i*. 

13.  X  -\-y,  x  —  y,  ;     ,   b,  a  —  b. 

14.  a*'  —  a*,  x^  +  «^  a?^  —  (i%  x  -\-  a. 

15.  of  —  (j-iu%  x^  —  KkiS  i-2  —  4rt2. 

16.  a-{-  b,  a"^  +  ^^/^  +  b'^,  a^  —  b*. 

Division  of  one  Polynoniiiil  by  another. 

If  the  dividend  and  divisor  ;irc  both  polynomials,  and  entire 
functions  of  the  same  symbol,  and  if  the  degree  of  the  numer- 
ator is  not  less  than  that  of  the  denominator,  a  division  may 
be  performed  and  a  remainder  obtained.  The  method  of 
dividing  is  similar  to  long  division  in  Arithmetic. 

96,  Case  I.  Wlicii  there  is  only  one  algebraic  syin- 
hoi  in  the  divide juI  mid  divisor. 

Let  us  perform  the  division, 

3ar*  —  4.f^  +  ^ix^  +  32'  —  1   ^  x^  —  x -\- 1. 

We  first  find  the  quotient  of  the  highest  term  of  the  divi- 
sor a^,  into  the  highest  term  of  the  dividend  ^x\  multiply  tiio 
whole  divisor  by  the  quotient  'dx/^,  and  subtract  the  product 
from  the  dividentl.  We  ivi)eat  the  process  on  the  remaindtr, 
and  continue  doing  so  until  the  remainder  has  no  power  of  / 
so  high  as  the  highest  term  of  tlie  divis(jr.  The  work  is  must 
conveniently  arranged  as  follows: 


•1 


^  ,V'  *  DiviH 


■<i  Fir-t  Uciiin 
i  -X  <  Uivir 


I   Scctiiid  Ucii 


Divis 


%  Third  and  li 

I        Tiie 
fi  hoeause 
ill  Arilh 
the   linn 
t  divisor. 


3/' 


■I        This 

'  by  I  he  d 

Thei 

I  respond  i 

ill  wliicl 

the  (lent 

,,  Aritlinu 

I  lUDider 

I  i  iuu  nia_ 

I  ]  I  roper  i 

Exec 
to  the  IV 

1.  11 

2.  D 


3-  1 

4.  E 

5.  I 


'a. 


ther. 

,  and  entire 
the  numer- 
visiou  may 
method  of 


raic  57/7? i- 


3f  the  divi- 
ultiply  the 
le  prodiier 
remainder, 
power  of  .." 
rk  is  must 


,3j'  X  DiviHor, 
Fir>t  lU'niainder, 
_,(    <  Divisor, 

Second  Kcmaiiider, 

.-•i  .;  Divifor, 

Thin!  luid  last  Koinaiiulcr, 


DIVIHION. 

Dividend. 

3.^4  _  4rJ  +  U^  +  3u-  —  1 

3.,-4  _  3.^3  ^  3.t;2 


6:3 


Divisor. 


X' 


^•  +  1 


-     X' 

+ 

yi  -\-  wx  - 

;/2          X 

i 

— 

tx^  +  4.6-  — 
2x^  4-  2^;  - 

•1 
3 

3x'^  —  X  —  'Z       Quolieut. 


2:c  +  1 


'I'hc  division  can  l)e  carried  no  farther  witliont  fractions, 
because  .r^  will  not  ^'o  into  .r.  We  now  ai)i)ly  tiie  same  ruh'  as 
ill  Arithmetic,  by  adding  to  the  (luotient  a  fraction  of  which 
the  numerator  is  the  remainder  and  tiie  denominator  tiie 
divisor.     The  result  is, 


3^  _  4.7,3  _|_  2a;2  +  3:<;  —  1 


=  3a;2  —  a;  —  2  + 


2.r4-l 
•/•2— ./•  +  ! 


K' 


This  result  may  now  be  proved  by  multiplying  the  <;;i(.  ieni 
by  ihe  divisor  and  adding  the  remainder. 

Tliere  is  an  analogy  between  the  result  {a)  and  the  cor- 
responding one  of  Arithmetic.  An  algebraic  fraction  1  o  {((), 
in  which  the  degree  of  the  numerator  is  greater  than  that  of 
the  denominator  may  be  called  an  improper  fraction.  As  in 
Arithmetic  an  improper  fraction  may  be  reduced  to  an  cnfire 
njonltcr  plus  a  proper  fraction,  so  in  Algebra  an  im})roper  frac- 
tion may  be  reduced  to  an  entire  function  of  a  symbol  i)lus  a 
])ro})er  fraction. 

EXERCISES. 

Execute  the  following  divisions,  and  reduce  the  quotients 
to  the  form  {a)  when  there  is  any  remainder. 

1.  Divide  .t-  —  2r  —  1  by  x  -\-  1. 

2.  Divide  .r^  +  'Zx-  —  tx  —  I  by  x  —  1. 

3.  Divide  .t^  —  3x^  +  Zx  —  1  by  a;^  —  x. 
2x^  —  2.^3  -|-jr2  —  X  —  5 

X^  —  X  —  1 


4.  Reduce 


5.  Divide  24^3  _  38^^2  _  33^  4.  50  by  2n 


A71S.  Quot.  =  12a^  —  a  —  - 


35 


-3. 

Hem.  =  — 


f| 


64  A/J.h'/ih'Aia    OrhVlATlOXS. 

6.   Divide  j'^  —  1  l)y  x—\. 

When  terms  lire  wanting  in  the  dividend,  tlicy  may  Im'  roiiHidiTcd  ;i> 
zero.  In  tliiw  liiHt  cxerciHf!,  tlio  trrniH  in  r\  a*-',  and  x  arc  wanting.  Hm 
tlie  beginner  may  write  the  dividend  and  i)erf()rni  the  operation  tliuH  : 

X*  +  0,r^  +  0/''  +  Ox  -  1   I  x  -  1 


I 


T* 


a^  +  x"^  +  X  +  1 


a^  +  0/^ 


ra- 


ce^ 


x'  +  Ox 
x'  —    X 

x^l 

x-1 

0      6 

Tlio  operation  is  tlms  assimilated  to  tliat  in  which  the  expression  is 
complete;  l)iit   tlie  aetual    writing  of  tlie  zero  terms  in  tliis  way  is  uii 
necessary,  and  sliould  Ik;  dispensed  with  as  soon  as  the  student  is  ahli- 
to  do  it. 

7.  Divide  a^  —  'h(  +  1   l)y  a  —  1. 

8.  Divide  x^  -\- I  by  x  +  1. 

9.  Divide  %a^  +  l;i5  by  'Za  -f  5. 

10.  Divide  a^  +  1   1)y  (t  +  1. 

11.  Divide  «'  +  'Za^  +  0  by  (fi  -|-  'Za  +  3. 

12.  Divide  «"  —  1  by  a^  +  'hfi  +  2a  +  1. 


13.  Divide  .t"  —  1  ■*./••  +  'Mkc^  —  3;2  by  x^  —  2. 


10. 


14.   Divide  (.r3  —  'Zx  +  \)  {.i^  —  I'Zx  —  10)  by  x^ 

For  some  purposes,  we  may  equally  well  perform  the  operation  by 
bejifinning  with  tlio  term  containinjj;  the  lowest  power  of  the  (juantity, 
or  not  containing  it  at  all.     Take,  for  instance,  Example  9  : 

125  +  ^n^     I  5  +  2a 

125_+  50a       25  -  10a  +  4a-' 

-50a 

-  50a  -  20a^ 


20'^-'  +  8a3 
20r/''  +  8a3 

15.  Divide  1  +  3.r  +  3;^-2  +  3^  by  1  +  x. 

16.  Divide  1  —  4.r  +  4.r^  —  .t^  by  1  —  x. 

17.  Divide  15  +  'Za  —  o(fi  +  (fi  -\-  Za^  —  w'  by  5  +  4«  —  (r. 

18.  Divide  1  —  if  by  1  +  2?/  +  2?/^  +  if. 

19.  Divide  04  — 04:?-  +  lG.v2— 8x•3+"4.^•4— .?:«by  — 4  +  2:r4-2-l 

20.  Divide  04  —  lG:/2  ^  .^  ^y  4  _  4,^.  ^  .^2. 


ImJs  ill  t 

Let  u 
10  power 

to  be  .r,  i 

Let   1 

the  divid 

dividend 

Let  1 
the  divis 
thiit  tiie 

Tiieii 

nec 
Iv  (livis 

1.  T 

ill  tlio  ' 

1VS])01U 

2.  T 

sliall  1)( 
tlie  (livi 

tlie  quot 

L  T 

iiiultiplj 

est  term 


2.  T 

plyinp;  t 
tiie  quot 


Rem, 

aeeordin 


DIVISION. 


05 


i.sidcrcd  ;i> 
tin^'.  I5iit 
\  tliuH  : 


)reH8ion  i> 
way  Ih  nil 
ent  is  ubli' 


-10. 

oration  l)y 
J  quantity , 


•  4rt  —  ((\ 
\-2x  +  x\ 


*.)7.  Cask  II.  IVhru  there  are  several  alj^chraic  syni- 
IhJ.s  in  the  ilirisnr  and  (tir'utciKl. 

Let  US  HU])])ose  thedivideiul  and  divisor  arniiiiiicd  acc(»rdin«T 
10  powers  of  some  one  of  the  symbols,  wliich  we  may  suppose 
to  be  .r,  as  in  J^  TG. 

Let  us  call  A  the  coetlieient  of  the  iiij^hest  power  of  w  in 
the  dividend,  and  //  the  term  independent  of  x,  so  that  the 
dividend  is  of  the  form 

^'1.'"  +  (terms  with  lower  ])owers  of.?')  +  //. 

TiOt  us  call  a  the  eoellieient  of  tbe  highest  power  of  w  \\\ 
I  lie  divisor,  and  h  tbo  term  of  the  divisor  independent  of  .r,  go 
that  tbe  divisor  is  of  the  form 

r/.r"*  +  (terms  with  lower  i)0\vcrs  of  .r)  -f  h. 

Then  we  have  tbe  following 

Tlieorem.  In  orch'r  that  the  diviclc'iKl  may  bo  exact- 
ly divisible  by  the  divisor,  it  is  necessary  : 

1.  That  tlie  term  containing  the  highest  power  of  x 
in  tlu^  dividend  shall  be  exactly  divisible  by  the  cor- 
resi)onding  term  of  the  divisor. 

2.  That  the  term  indopendont  of  x  in  the  divid(Mid 
sliall  be  exactly  divisi])l(i  by  the  corresi)onding  term  of 
the  divisor. 

Rcamn.  Tbe  reason  of  this  theorem  is  that  if  wo  suppose 
tlio  quotient  also  arranged  according  to  the  powers  of  .r,  then, 

1.  Tbe  highest  term  of  the  dividend,  Ax?^^  will  be  given  by 
nniltiplying  tbe  highest  term  of  the  divisor,  ra-*",  by  the  high- 
est lerm  of  the  (piotient.     Ilenco  we  must  have. 

Highest  term  of  quotient  = 

2.  The  lowest  term  of  tbe  dividend  will  l)e  given  by  multi- 
plying the  lowest  term  of  the  dividend  by  the  lowest  term  of 
tlie  rpiotient.     Hence,  we  must  have. 


Lowest  term  of  cpiotient  = 


H 

h 


Rem.  1.    Since  we  may  arrange  tbe  dividend  and  divisor 
according  to  the  powers  of  any  one  of  the  symbols,  the  above 


f 


00 


A  UJ FnilA  re   OPFRA  TTONS. 


thooroni  must  bo  true  wlmtevor  8}'nibol  wo  tiiko  in  [\\v  place 

of  J'. 

Rkm.  '?.  It  (looH  not  follow  tliat  uli(>n  tho  conditions  «»f 
the  tlu'oivnj  are  riillilU'd,  the  division  can  always  be  performed, 
'ri.'is  (|uef>tion  can  \)v  decided  only  by  trial. 

Wc  uow  reaeh  the  followin*^  rule: 

\.  .  Imtinjc  hotli  diridrnd  (tiid  di visor  (ir.cord'nt'^  lo 
ihc  (tsrciidiifj^  or  descending  jxnrers  of  sonic  comnKm 
St/ in  ho/. 

II.  Forn/  the  prst  term  of  the  (jnotient  hf/  diridhrj 
the  /irst  term  of  the  diridend  Inj  the  Jirst  term  of  the 
dii'isor. 

in.  Afnlff/)7f/  the  lehole  dirisor  hif  the  term  thus 
foinni,  iind  sulitroet  the  jn'oduct  j'nnn  the  dirhtend. 

IV.  Treat  the  reniffinder  rts  o,  iieir  diridend  In  the 
Sffnie  ii'fui,  and  repeat  the  process  an  til  a  rcniftliidcr  is 
found  irltlch  Is  not  dlrlslIjJe  hr/  the  r/aotlent. 

Ex.  I.     Divide  x^  -\-  'Sax^  +  oah:  -f  «='  by  x  +  a. 


OfEHATlON. 


2ax^  -\-  'Sale 
2ax^  +  2uh: 


a\c  4-  a^ 
a^x  4-  a^ 


0        0 
Ex.  y.     Divide  x^  —  ax^  +  a  {b -^c)x  —  alc—hx^—cx^-i-Ijcr 
by  2-  —  <(, 

Arranging  acconMnff  to  §  76,  we  have  the  dividend  as  follows: 
•'■^  —  (f(-{-b-\-  c)  x'  +  {(d)  +  be  +  c(f)  X  —  abo  \  x  —  (t 

x^—  (b-\-c)x  -{-  he 


x^ 


ax^ 


—  (/;  +  c)  .r2  4-  {ab  +  be  +  ca)  x 

—  {b  +  c):}"^-^  {ab-\-ac)x 


bcx  —  (the 
hex  —  abc 

0  0~ 


i 


I. 

2. 

3- 
4- 

5- 

s. 

10. 

1 1. 


FiiAcrroNs. 


<;7 


EXERCISES. 


I. 
2. 

3- 
4- 

5. 


Div 
Div 
Div 
Div 
Div 

6.  Div 

7.  Div 
S.    Di\ 

9.   Div 
10.   Div 


II. 

12. 


Div 

Div 
Div 


ido 
ifk' 
ido 
ido 
ido 
ido 
ide 
ide 

id(! 
ido 
ido 

ido 
ido 


tlio  dividond  of  Kx.  2  ultovo  l)y  j'  —  h. 
tho  dividend  of  Kx.  2  above  l>y  ./■  —  r. 


(!•' 


^  />:«  -f.  Wah  —  1    |)y  a  J^  h—  1. 


(t^li^  +  'iahyi  —  {(('^  f  l)i)x^  by  ^^Z*  +  (^/  —  //)  .r. 
(r,2  _  hr)^  -f  S//V-1  l)y  ^t'J  +  lir. 
[a  -f  />  -f-  r)  {ah  4-  /yr  +  m)  —  ahr  by  ^/  -f  h. 
\((  j^  b  —  v)  {h  -¥  c  —  a)  {r  -f  a  —  h) 

l)y  (fl  —  Ir  —  r^  ■\-  'ihr. 
f(^  -^  /;'  4.  r''  _  3^//*r'  by  ^^  4-  /;  -f-  c. 
./ «  +  4«»  l)y  :/•«  —  2r/.r  -f-  2a-. 
a"^  {b  -\-x)  ^  1/  {x  —  «)+(«  —  h)  x^  4-  «i^ 

by  X  -\-  a  -\-  U 
:,-3  _  ^,.r2  _  hh-  4-  ^//r'  l)y  (x  —  «)  (.r  4-  //). 

UaKc^  —  i\a\i^  4-  Oi^Ki'^  —  a'  bv  :>/f2,^a  —  a\ 


-♦-♦♦- 


CHAPTER     IV. 
OF    ALGEBRAIC     FRACTIONS. 

OS.  Def.  All  Algebraic  Fraction  is  tho  (expression 
of  Jin  indicated  quotient  when  the  divisor  will  not  ex- 
actly divide  the  dividend. 

Example.     The  quotient  of  ;;  4-  q  is  the  fraction   -• 

Def.  The  numerator  and  denominator  of  a  frac- 
tion are  caUed  its  two  Terms. 


Tr.ansforniatioii  of  Sinj»:le  Fractions. 

1)9.  Reduction  to  Loioesl  Terms.  If  the  two  terms 
of  a  fraction  are  multiplied  or  divided  by  the  same 
(inantity,  the  value  of  the  fraction  will  not  be  altered. 


I 


68 


A  L  G  EBRAIC   OPERA  TIOXS. 


ExAMPLi;.     Consider  th(^  fniction 


ax 


ay 

X 


If  wc  divide  both 


terms  by  a,  the  fraction  will  become  - 

J  y 

ax  _  X 

'^y  ~  y 

Corollary.  If  the  niinierator  and  donomiiiator  con- 
tain common  factors,  they  may  be  cancelled. 

D(f.  When  all  the  factors  common  to  the  two 
terms  of  a  fraction  are  cancelled,  the  fraction  is  said  to 
be  reduced  to  its  Lowest  Terms. 

To  rcdncc  a  fraction  to  its  Joircst  terms,  foctor  both 
terms,  when  necessary,  and  cancel  all  the  coinnwn, 
factors. 

Ex.  I.     — -^  =  -. 

acny*       en 

The  factor  ay^  common  to  both  terms  is  cancelled. 


Ex.  2. 


a'W' 


«= 


The  factor  aW  common  to  both  terms  is  cancelled. 


a'^x 


Ex.  X.     Keduce  -V 
a^x 

Here  ({"x  is  a  divisor  of  both  terms  of  the  fraction.     Di- 


viding by  it,  the  result  is    .j*     Ilcncc 

(6 


(Ox 


1 


a*- 


E 


x.  4. 


Ex.  5. 


a'i  -(-  ^al  -^  11  _    {n_±hY «  + 


0 


a'-  -  b^ 


{a  -f-  h)  (a  —  b)        a  —  b 


w  a  —  nu  _  {in  —  n )  u  _ 


mx  —  nx 


{in  —  u)  x 


u 

—  a 

X 


%. 
J) 


I  I. 


13- 


>.^- 


EXERCISES, 


Reduce  the  following  fractions  to  their  lowest  terms 


lOpqr 


ayn 
ahnx 
I2axy 


idc  hotl 


tor  con- 

lio  two 
said  to 

'tor  hotli 
loninwii 


on.     Di- 


IS : 


t 


FRACTIONS. 


69 


30  {(i^  —  'Zax  -\-  .c2) 
(ijl  —  by 

— ■  ■  ■■  -     -  • 

bx 


ax 


U'  -  Z>2 


ai  _  'lab  4-  -^ 
01^  -f  y^ 


I  I  .  —7 


13- 


(f* 


b* 


17- 


a-  -  //-^ 

,,.:,   __   y!) 

axui  —  axn 

_ —    - .  —  , 

/y ///;/,  —  byn 


6. 
8. 

10. 
12. 
14. 
16. 
18. 


:20(^r  4-  .t)(?/?  —  w) 


(l^f^-^bhl' 

ay  — by  ' 
r/2  +  Aax  +  4j2 

rt2^4arJ 
r/3  4-  8/y3 


a^  -\-jib  4-^ 

)nx  —  nx 

\a  -\-  b)  (m  —  n) 


lOO.  Rule  of  Siynx  i)i  Frartioiix.  Since  a  fraction  i.<  an 
indicated  (|U()tiont,  tiiu  rule  of  signs  corre.s])ond.s  to  that  for 
division.  Tlie  following  theorems  follow  from  the  laws  of 
]mdtiplicatlon  and  division: 

1.  If  the  tenns  are  of  the  same  sign,  the  fraction  is 
positive ;  if  of  oi)posite  signs,  it  is  negative. 

2.  Changing  tlie  sign  of  either  term  changes  the 
sign  of  the  fraction. 

8.  Changing  the  signs  of  hoth  terms  leaves  the  frac- 
tion with  its  original  sign. 

4.  The  sign  of  the  fraction  may  he  changed  by 
changing  the  sign  written  before  it. 

n.  To  these  may  be  added  the  general  principle  that 
an  even  number  of  changes  of  sign  restores  tlie  fraction 
to  its  original  sign. 


Ex. 


a        —  a 


a 


I.      .  ■= 


h        -b 


,^  a  —a         —a 


Ex.  3 


a  —  h 

m  —  n 


b_-a 


a 

-b 

a 

" 

-  b 

a  ■ 

-b 

n  - 

-  m 

b  —  a 


^    ( 


I 

V 

]        I 


in 


n 


70 


A LGEIillA  10   OPERA  TIONS. 


T 
( 

i 


^ 


■9 

i 


EXERCISES. 

Express  (lie  following  t'ruetiuns  in  Coiir  din'crent  ways  with 
respect  to  si^nis: 

X  —  V 

I .      '^ . 

a 

m 


V 
5-     - 


II 


2. 

4- 
6. 

X  —  y 

7r-b\ , 

a 

((  +  ///.  —  X 

7. 

9- 
1 1. 


-h 


(t  +  ./;  —  // 

a  —  X  ^  h 

X  —  n  -\-  h 

'  h-x 


10. 


p  -\-  I]  —  r  a  —  '///,  4-  X 

Wrile  I  he  followiui;  friiftions  so  that  the  symbols  a;  Jind  y 
sliall  iic  positive  in  both  terms  : 

./•  ~  h  „  m  —  X 

+  .  8.      + 

n  —  II 

a  —  .6' 

/;  —  X 
<(  -\-  h  —  X 
a  —  h  -\-  y 

101.  Whni  tlic  innTKT.'itor  is  Ji  ] )r(Kl net,  any  ono  or 

more  of  its  factors  can  be  r(Miiov<'d  tVoiu  tlie  miniciator 

and  inaTlc  a  nmltiplier. 

ahmx  ,     mx 

-  —  ah 

V  +  y  V  ^  'I 


K} 


'.X. 


X  1 

~  abm =  abmx 


V  +  7 


V  ^n 


EXERCISES. 


Express  the  following  fractions  in  as  many  forms  as  possi- 
ble wit.h  respect  to  factors  : 


pqx 
mn 


ah 


J.4    _ 


4.  — 

a 


-b 


(C 


-  // 


nbc 

a~^b' 

x  -\-  'la 


MYt,  Redvctfon  io  Oh^eii  Drnominator.  A  quan- 
tity may  b(.»  ('X])n'ssed  as  a  fitiction  with  any  rocinircd 
denomiriator,  />,  by  supixisinu'  it  to  have  tlw  j'lMionii- 
Tiator  1,  and  thru  miillii)lylii<2;  both  terms  b>   D. 


(I 


For,  if  we  call  a  tlie  (plant  ity,  we  have    a  ~      = 


II 


D' 


Ex. 


If 

\        tVa('ti< 

])f()(lu 


*      both  111 


4- 

5- 
6. 


FRACTIONS. 


71 


1)0S.<1- 


Ex.  If  wo  wish  to  express  tlic  qnaiitity  «A  as  a  fraction 
hiiviug  xy  for  its  denominator,  wo  write 

ah.iif 
xy  ' 

If  tli('  (quantity  is  fiactioiuil,  hotli  terms  of  tiie 
tVaetioii  must  be  multiplied  hy  that  factor  wliich  will 
])i(Kluee  the  rtn|uii<'d  denominator. 

lv\.     To  express       with  tiie  denominator  nlfl,  we  niultijily 

holli  nieinhers  hy  nlr^ -^  h  =  nl?.     Tiuis, 

a  _  a)ib^ 

b  ~    nU^  ' 

This  [:'   cess  is  the  reverse  of  reduction  to  lowest  terms. 

EXERCISES. 

Exj)ress  the  quantity 

1.     a  Avith  the  denominator  h. 


J- 

4- 

5- 
6. 

7- 
8. 


ax 

ab 

ni 

n 

-  1 

in  {n  —  p) 

X  -\-  y 

X  —  II 
X    +  1 

a  —  i 


a 


u 


a  (( 


ti         a 


it 
it 
u 
« 

« 

« 


ax. 

ab'K 

n  {x  -  y), 

X. 

a2  _  bK 
u^  -  f. 
.r2  +  2a;  +  1. 
r/«  -  1. 


Noj»a(r.>»  Exi>()ii(»nts. 
lOo.    By  tiie  principle  of  j^  85,  we  ha\e 

((a 


(t>^ 


=  rt"~*. 


If  we  have  k  >  n,  the  exponent  of  the  second  member  of 
tiie  e(|nation  will   •"  .u'<j;ativ«',  and  the  first  meml)er,  hy  can- 


72 


A  L  a  PJIiltATC   OP  ERA  TIONS. 


i    '^' 


1  i 


celling  n  factors  from  each  term  of  the  fraction,  will  become 


(V 


k-n 


Hence 


a 


k-n 


^  —  ««-*. 


—  a 


By  i)'itting  for  shortness  k  —  7l  =  .v,  the  equation  will  be 

i  =  a-', 
a" 

Hence, 

,4  nc^ativG  cx])nncnt  iiu/icdtcs  the  recipvocdl  of  tJir 
coJ'resjwfKling  (jiiaiitlty  irith  n  positive  e.r]>oneiht. 

ffH 

If  in  the  formula  «""*  =    .    we   suppose  k  =  w,  it  will 


(('^ 


a^ 


become  «"  =  -,  or  ^^  —  [.     Hence,   because  a   may  be  any 
(|uantity  whatever, 

,/////  f/fiajhtiti/  irith  the  rxponeiit  0  /.s'  f(inifl  to  iniitif. 

This  result  nuiy  l)o  made  more  cloar  })y  suc- 
cessive divisions  of  a  ]»ovver  of  a  h\  a.  Every 
time  wo  clfcct.  tliis  division,  we  diminish  the  rx- 
ponent  hy  t,  and  we  may  PU])pos('  this  diniiniition 
to  contimuf  al^el)raically  to  negative  values  of 
the  exponent.  On  the  left-hand  side  of  the 
equations  in  the  margin,  the  division  is  etlected 
symholienlly  l)y  diminishing  the  exponents  ;  on 
the  right  the  result  is  written  out  in  the  usual 
way. 

E  X  E  R  C  J  '"-  K  S . 

In  the  following  exc^'cisep.  urii"  'he  quotients  which  arc 
fractional  both  as  fractions  reduced  to  their  lowest  terms,  am! 
as  entire  ((uantities  with  negative  exponent -;,  on  the  principle 


r/3 

: — 

aaa 

d^ 

— : 

aa 

r/i 



'!■ 

r/O 

's:.— 

1 

rt-i 

-— 

1 

a 

r?-2 

-■ 

1 

cic. 

etc. 

a 
h 


itb-\ 


a%~^,     etc. 


Divide 

1.  7^  by  a;. 

2.  X  hy  x^. 

3.  _  2Z»3  by  t)\ 

4.  \ah^  bv  —  'laH). 


Ans.  ,r. 


1 

X 


Ans.  '  or  x~* 


A  ns. 


n'i 


a^ 


or  —  ^n-W. 


FRACTIONS. 


73 


II  become 


ti  "will  be 


a 
t. 

of 

tlir 

n, 

it 

will 

y 

be 

any 

)  Kumj. 

=     ((((a 
—     aa 

z-       .1 
I 

a 
1 

ltd 
eve. 

*vl)i('h  arc 
erms,  ;iii(i 
principle. 


Ans.  r. 


or  x~^. 


■  ^n-%\ 


10 

1 1 

12 

14 

i6 

17 

1 8 


—  ^a^b  by  4r//y2 


6.     l'in^\r>/  by  4rtZ».i\ 


l-lrr'^V-a  by  —  :'aV/ci         8.     'Uapq.ry  by  Iba^r, 
4Srt-^  (.1-  -  ^)-^  by  ;3(i  (.r  -  //). 


A'llr 


^3'''<^") 


"i'l  {a  —  b)  {m  —  u)  by  15  {a  +  z^*)  (/m  +  n), 

^^^,  (^<2  _  ^^2)  (^^^3  _  y^2)    by    15  (.^  _  ^,)  (^,;   ^  ,,^)^ 
(,;.4  _    1)  i^ai  _  4^2)     by    (^p2  _  1)  (^a  _^    2/>). 

./:<■>—  1  bv  .r  4-  1. 
rtV;3, ■•«//■'  by  a^h\rhi\ 
m^'n^ifz  by  mii^ii^f. 

m  {m  +  1 )  (/yz  + 1>)  (/«  + .'})  by  m  {m  —  1 )  {m  —2)  (m  — 3). 
19.     a"^  by  «".  20.     rt6'"'6'''  by  qb'^c'"K 

Dissection  of  Fractions. 

104.  If  the  iminei'Jitor  is  a  polynomial,  cacli  of  its 
terms  may  be  divided  separately  by  the  dero.^Tiinator, 
and  the  several  fractions  connected  by  the  signs  +  or  — . 

The  principle  is  that  on  which  the  division  of  i)<)lyn()niials 
is  founded  (J5  8?).     The  general  form  is 


A  +  B  ->rC  -{-  eti;.        A       B       C 

— Z T — T. =  --  +  -  H \.  etc. 

m  m       ))i       III 


(1) 


The  separate  fractions  may  then  be  reduced  to  their  lowest 
terms. 

Example.     Dissect  the  fraction 

3i)nra/vV_  iSrm///  +  1  '^bnz  —  VWhfyi 

The  general  form  (1)  gives  for  the  separate  fractions, 

\yi(^iWx       \Saini/       Uibnz        Vi/ihihf 
i^abx  ~  IJUih-        U\ab.i         y\ab7 ' 
Reducing  eai-h  fraction  to  its  lowest  terms,  the  sum  becomes 
,       9iny      15«z       'dbn^ii 
""  "S^i  '^'  Wax  ~  '4:ax  ' 


f 


i 


74 


ALG KBRA IC   OPERA  TIONS. 


.   .r 


iF't 


E  ^;  n:  R  c  I  s  E  s . 
St'paratc  into  sums  of  fractions, 
ahc  -\-  bed  +  cda  -\-  dab 

((bed 
—  xj/zd  +  x^yz}(^  +  xijh'U  —  x-if^z^u^ 

(("'  -  b^  ■  ((h-  -  /ry 

3-       -  „A—  4. 


2. 


5- 
6. 

7- 


(y//  —  i()  {n  +  y)  —  i>'f  +  >0  (;>  —  v)^ 

{m  —  n)  {/)  —  (/) 
(■'■  -  ^0  (//  -  ^>')  +  {•>'  -  !/)  {(f  -  b)  +  {x  -  h)  (//  -  a) 

rr'^  —  !l^ 
{a  +  />>)  (w  —  n)  —  {a  —  b)  {m  -\-  n) 
'r/'-"^-  b^ 


Ajy^'ojyatioii  of*  Fi'ju'tioiis. 

105,  When  several  fractions  liave  eqnal  donomina 
tors,  their  sura  may  be  expressed  as  a  single  fraction 
by  a<]^gregating  tlit^ir  nnmei'ators  and  writing  the  com- 
mon denominator  under  them, 
A 


Ex.  I 
Ex. 


m 


B      CI  _  A  -  n  +_c 

m       m  ~  m 


a  ~  h       b  —  c        c  —  (I 

2.     H H 

X  —  y      y  —  X      X  —  y 

X  —  7     -^  —  y     ^  —  y  ~~  ^  — .'/        -''  —  // 

Rem.    This  process  is  the  reverse  of  that  of  dissecting  a  fraction. 

EXERCISES. 

Aggregate 

a        ab       abc-  a 

abc      abc      abc 

"x'—  ff-    ,y  —  ^      a  -\-  b      x  —  y 
((\z  ah'  a^x  a^x 


(a  -  bf       {tt  -  by 


3- 


a  —  b      b  —  a 


a 

III 


c  d 

a  —  b       b  —  a 
a  —  c        c 


n      m 


b        c  -\-  a 
n      n  —  m      n  —  m 


^) 


lomina- 
Taction 
le  com- 


FliACTKhVS. 


7,") 


100.  Wlioii  all  llu'  fractions  iiavc  tiot  tht-  same  tlcnoiniiia- 
tor,  IIk'}'  niiist  be  rt'duccd  to  u  common  denominator  l»y  the 
process  of  §  102. 

Anv  common  nuill )[)!('  of  the  denominators  may  i)e  taken 
;i-  the  common  dcnomiualor,  hut  the  least  common  multiple  is 
till'  .<inii)lest. 

To     UKDUCK     TO     A     COMMON      DeNOM  IN  ATOi;.       C/lonsr    f(, 

colli  III  itii  III  nil }  pic  of  the  (Iciioni'nuitdi'K. 

,]//illi /ill/  htitli  Icrnis  oj'  (t/r/i  fntctlnn  hif  l/ie  iiiiilti- 
plii'i'  ncccssKrij  to  cliaiiga  its  dcnoini iidtor  In  Ihc  vliosrii 
III  lilt } pie. 

Notf:  1.  ^riie  rei|nired  multipliers  will  he  the  ([uotients  of 
I  he  chosen  multiple  by  the  denominator  of  each  septirate 
fraction. 

KoTt:  9..  ^Vhcn  the  denominators  have  no  commcm  fac- 
tors, the  multi))lier  for  each  fraction  will  he  the  product  of  the 
denominators  of  all  the  other  fractions. 

XoTK  3.  An  entire  (puintity  must  bo  regarded  as  having 
the  dcnonunator  1.     {%  lO-.'.) 


I 


( 


EXAMPLES. 


—  y 

ction. 


by 


I.     Aggregate  the  snm 

111  1 

1 1 J. 

a       ab       abc       abed 
in  a  single  fraction. 

The  least  common  multiple  of  the  denominators  is  abcil. 
The  separate  multii)liers  necessary  to  reduce  to  this  com- 
mon denominator  are 

abed,     bed,     cd,     d,     1. 

The  fractions  reduced  to  tlu'  common  denominator  ^/i^'i/  arc 

abed       —  bi-d       -f-  rd       —  d 

abed'       abed''     abed'    i^^S^*^bcjiL*u^, 

abed  —  bed  4- 

ab(^  . 

Hy  disse(!liiig  this  fraction  Ks  in  ><  iBftHrTiiay  15c  reduced 
to  its  original  lorm. 


The  sum  is 


t 


70 


A  L  G  I'JIiltA  IC   OP  Ell  A  TIONS. 


I 


2.  llcduce  the  sum 


to  a  siiiKlu  fraction. 


1      a      b      G 
a      b      c"  d 


Tlu'  mnltii>liors  are.  ))y  Note  2,  bed,  acd,  nbd,  abc. 

Using  tlicsc  niultiitliery,  the  fractions  l)ccome 

bed       —  d'cd      a/j^d       —  abc^ 
abcir       abed  '     aberr       abed  ' 

from  which  tlic  required  sum  is  readily  formed. 
3.  Kcducc  the  sum 


1  X 

^  .r  -  1  ^  :/;  +  1 


Tlie  least  common  multiple  of  the  denominators  is  x^ 

The  multipliers  are,  by  Note  1, 

x^  —  1,     X  ■{- 1,     X  —  1,     1. 

The  sum  of  the  fractions  is  found  to  he 

x^  —  1  +  X  -\-  1  -^  x^  —  X  -\-  a^  _     3x^ 

-.2  _  1  —  ^_~i* 


—  1. 


EXEF^CISES. 

Reduce  to  a  single  fraction  the  sums, 


I 

3 

5 

7 

9 
10 

II 


1    u 


X—  1 

1  1 


2.       1  — 


1  —  X         1  -^  X 


X 


ax 


x" 


a  -\-  X      a  -i-  X 
a  X 


4 
6. 


X  {a  —  x)       a  {a  —  x) 
1  2?/ 


X  -{-  }j       :?-^  —  y/2       :.—  If 

1  1  1 

a  —  b      b  —  c      c  —  a 


a  a 


X  +  y      x-y 


12. 


X  -\-l 


1  1 

+ 


1  —X         1  -{-  X 

a  b 

a  —  b      a  -\-  b 
5 


2x-.b 

•  8-  4^2Tn  + 


2a; -1 


3 

X 


a  -\-  b      a—  b 

. — ,   „  I.,         —  « 

a  —  b      a  ■\-  b 


1 


a;2—  1. 


3 

—  a 

X 


13- 
14. 

15- 
16. 

iS. 

19. 

20. 
21. 

22. 

23- 
24. 

25- 
26. 

27. 
28. 

29. 

30- 


-  ( 
1 


FRACTIONS. 


a 


Tl 


ai  _  //a      rt  —  ^  "^  r/  +  6' 


a  (a;  -  1)       'Z  {x  +  1)      a;2 
rt  _  ^       \         a  —  bl 


1-      « 


17.   y      '"  -^  y 

m'^      m  (w  —  y) 


■yi 


a 


a  —  X      d^  —  x^ 


/>        Z»  —  C        6"  —  r/.        (r^  —  //)  {h  —  r)  [c  —  a) 
a'+O       b~^~c       c~^7t  "^  (^+  /*)  (/>  +  r)l;c^rt)* 


m  —  (x  —  a)      7)1  —  (./•  4-  a) 
X  ^  y        ~~        X  —  y 


r?^       ^c       «c 
a 


(^^  _  b)  (^,  _  "^  +  {h  _  ^^)  (<r^^  "^  (c  —  r/)  (6-  -T)' 
a;  +  1       :?;  —  1 


x—V       a;  +  1 

a  X 


X  -\-  a      X  —  a 
■2 


x^  —  2a;?/  +  ?/- 
a;2  -f-  ?/2 


/y^^ 


«2    _|_    ^y2 
J^  1 1 

"(rt  +  Z*)'^  "^  {a  -  bf  ^  d^  -  6« 


r  I 


. 


78 


A  UU'UniA  l( '   OPHUA  TIONS. 


¥i\v{in'\\\)x  FriK'tioiis. 
I()7.  Tf  scvcrjil  terms  of  the  immenitor  ronlaiii  a 


roiuiiioii  ra('t( 


th 


'fncieiits  of  this  WivU 


I) 


•oiuiiioii  i.'icioi-,  tiK'  coenicKMiTs  or  tins  ijictor  tikiv  ne 
.'Hhh'd,  jiiid  tlicir  a«i;i;i'('<i'{it('  iuiilti])li('(l  hy  the  factor  for 
a  new  form  of  the  inimeratoi". 


I. 


EXAMPLES. 

ii.r  —  hx  -|-  c.r  +  dx  __  {a  —  h  -\-  c  ^  d)  x 


m 


m 


X 


=  {a-h-\-c  +  (/)-'     (§101.) 


in 

nhx  -f  hex  4-  (icji  —  ahjj  _  {dh  -f  hr)  x       (ac  —  n/))  // 
((On  ((On  fffni 

=  {n  +  c)  •'-  +  (c  -  /.)  .^. 


Kc'ilncc 


EXERCISES, 


a/)f/  —  he II  —  acy  mnn,  +  ?»yt;?^  -\-  pnu 

aba  '  mu 


3- 

4- 

5- 
6. 

8. 


10. 


aba 
ax  —  /;//  —  3Z>.r  —  4^/?/ 

"Zma 
Amx  -f  2//  —  ^ax  —  Ora;  -f  rry 

xyz 
a^  4-  2rt2^»  +  al)^ 


a^x  —  4aZ'c  —  (3//  —  4r)  ^« 


.T//  J)  +  y 

:r3?/  —  [4./-  +  X  (2b  —  -U;)  -j-  .^r/r] 

r^r.?'^  —  4ra:  —  3  [w./:  +  ?m  («  —  x)  —  atn] 


2a  —  U 


iaVx  —  2cVx  +  2bVj'  —  2  {mn'^x  —  4\/:r), 


(ilor 
lie  II  I) 


3rt  —  \h 


FHACT/OAS. 


70 


?/ 

— « 

On 


MuUil>1i<'ii(i(>ii  aiKl  Division  oC  FnicrMMis. 

|(),S.  FnufhtntiiUal  Thcon  in.s  in  the  M  filfipliciUion. 
and  Diri.siou  of  Fractions  : 

Thi'orciii  I,  A  fVnction  may  !)(»  iriulti]tli(Ml  hy  any 
.'liuiiitity  l)y  citlicr  iiniltiplyiiiL;-  its  niiiiu'ratordrdix  idiiio- 
its  (ieiKUiiinator  by  that  ([uantity. 

Cor.  1.  A  fnictioii  iiiiiy  ho  mullipiit'd  hy  its  (li'Moiniiiutur 
l.y  simply  cjiiuH'lhii^^  it. 

('or. 'i.  ir  the  (It'iioiuiiiutor  of  the  iVaction  is  u  factor  in 
till'  iiiiiltipiicr,  cancH'l  tiu'  ili'iioniiiuitor  to  iiiiiUiply  hy  liii.s 
factor,  and  tiien  inidtii)ly  the  miiiu'rator  hy  this  otliiT  factors. 


Ex. 


7)1 


X  (ii  (,ci  —  l)^)  —  am  (./:  -f  />), 


a  [x  —  b) 
hccausc  tlic  imdtiplicr  r/-'  (.<:'  —  /A')  =  a  (x  —  Ij)  n  (./;  +  V). 

Theorem  II.  A  fraction  niay^  Ix'  divided  by  (dthcr 
dividing  its  nnnu'rator  or  niuiti])]yin,i2:  its  dcnondnator. 

Theorem  III  To  multiply  by  a  fraction,  the  multi- 
plicand must  be  multi})licd  hy  the  numerator  of  the 
fraction,  and  this  ])roduct  must  be  divided  by  its  de- 

liOMUUatol'. 

Let  us  multinlv    ,   liv  — 

We  multiply  hy  ///  hy  multiplying^  the  numerator  (Th.  I), 
and  we  divide  hy  n  hy  multii»lying  the  denonunator  (Th.  II). 

Hence  tlie  ])ro(luct  is  ,    • 

bn 

That  is,  fhc  product  of  the  miDicrnfors  is  the  nmner- 
(itor  of  the  ir(/iiirc(l  frortion,  (did  tlic  /ji'odnct  of  f/ia 
daiioiniiuttors  is  its  dciioniiiuitor. 


Multiply 

I. hy  :>• 


EXERCISES. 


X  —  a 
ah 


—  a. 


—  X 


hy  xy. 


ah  ,      X 

2.        -       OV       • 

X      '^   a 

ac     . 

4.     hy  X 

^      x  —  a^ 


■2 


a\ 


■  I 


IMAGE  EVALUATION 
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1.0     l^  iW  IIIIIM 


M 

32 


I.I 


m 

■  40 


M 
M 

2.0 


1-25  i  1.4 


1.6 


Photographic 

Sciences 
Corporation 


y 


A 


«j      M-i    ^      ///„ 


C^^^ 


iV 


N> 


^V 


#^^2^*^- 


6^ 


23  WEST  MAIN  STREET 

WEBSTER,  NY.  14580 

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80 


A L GEBRAIC   OPERA  TIONS. 


I'- 


i 


i     *, 


aim 
a  —  h 


by  mf 


m 


m 


by 


a  +  b 


ill 


6.     -^-  by  rta;3 
8.     a  + 


??i  —  a 

• — '       ( 

X  —  m 


m 
]i 


n 


X 


9.     ab  -  ^  ))y  ay  + 


If  — ah 


X 


10. 


???  +  ?i 


III  —  n 


-  by 


oy  ?i  H 

n  —  111 
m  +  ?i 


/>.c  ,      a 


II.     Multiply  r«  +  --  by  y  +  -  + 


?yi 


« 


12. 


Ilea  ace  |  in  H 1   in 1 . 

V         in  —  n!  \         m  +  nl 


13.  Reduce  ia \ih ~\ 

14.  Multiple  b ~  by  -• 

^  -^  a     ''  X 

711 

15.  Divide  —  by  p. 

16.  Divide  7  hy  a  4-  h. 

a  —  0    '' 


Ans.  — 
np 


17. 


Divide  _  by  re  —  1. 


a;  +  1 


-7 


18.  Divide  4-^-1  by  1  +  x\ 

X    —  L 

TA .  1 T     ■■""  r^a  —  oin  ,      -, 

19.  Divide  ---^-j-^^^-  by  i'*  -  fl^'^. 

109.  Reciprocal  of  a  Fraction,    The  reciprocal  of 
a  fraction  is  formed  by  simijly  inverting  its  terms. 

For,  let  ^  be  the  fraction.     By  definition,  its  reciprocal 
will  be 

a 
b 

Multiplying  both  terms  by  b,  the  numerator  will  be  b  and 
the  denominator  -  x  b,  that  is,  a. 

TT  h 

Hence  the  reciprocal  required  will  be  -,  or,  in  algebraic 
language,  ^ 


I. 


FRACTION'S. 


81 


a 
b 


a 


110.  Def.     A  Complex  Fraction  is  one  of  vvliich 

eitlier  of  the  terms  is  itself  fnictional. 

a 
h 


Example. 


X 

m  A — 

y 


((  X 

is  a  complex  fraction,  of  which     is  the  numerator,  and  m  + 
tlio  denominator.  ^ 

The  terms  of  the  lesser  fractions  which  enter  into  the 
numerator  and  denominator  of  the  main  fraction  may 
he  called  Minor  Terms. 

Thus,  b  and  y  are  minor  denominators,  and  a  and  x  are 
minor  numerators. 

To  reduce  a  complex  fraction  to  a  simple  one,  mnlti- 
phj  both  terms  by  a  multiple  of  the  minor  denominators. 


Example.    Reduce 


am 


y. 


b       h 

y'^  X 


Multijilying  hoth  terms  by  xy"^,  the  result  will  be 

amx 

winch  IS  a  simple  iraction. 

EXERCISES. 

Reduce  to  simple  fractions  : 


I. 


1+5 

y 


1 

a 

X 

'~y 

—  X 

a 

-hx 

a 

4- a; 

2. 


X 


a  —  X 
6 


a  — 

ab 

mn 

~W 

km 


b 

X 


;   i 


H  il 


«  'f 


.'[ 


m  in 


82 


ALU  EBUAIiJ   OP  Eli  A  TI0N8. 


I     f*". 


*. 


1^      'T 


■•il  ■ 


IT. 


13- 


1  + 


n  —  l 

il  -\-  1 


1  - 


n  —  l 

n  +  1 


am  - 

'■Vn 

an  - 

1  + 

h 
n 

{u  -  bY 
Aab 

1  - 

b^-a^ 

'  '  2ab 

1 

a 

a 
a 

+  a 

-}-2b       a 
-\-b  '^  b 

b 

2b          a 
a~'-\-'b 

6. 


8. 


10. 


14. 


1  +  a*      1  —X 
T~—x  "*"  Y'+x 


1  +  X 
1  —  X 

2x  — 


\—x 

1  + :« 


2/ 


a  -\-  b  — 


a 


12.     - 


1 

-\-a 

r 

1 

a 

1 

1 

a 

1 

—  a 
*3  ^ 

1  + 

a 

b 

X 

1 

+ 

1 

a 

!/  + 

X 

X 

+  // 

x^ 

X 

-H// 

r2     - 

f 

X  —  XJ 


X* 


y 


■i  I 


Division  of  one  Fraction  by  Another. 

111.  Lot  us  divide  j  by  —    The  result  will  be  expressed 

by  the  complex  fraction 

a 

m 
n 

Reducing  this  fraction  by  the  rule  of  §  110,  it  becomes 

an 
bm' 


Vrhich  is  equal  to 


Y  X  —        That  is, 
b       m 


To  divide  by  a  fraction,  we  have  only  to  multiply  by 
its  reciprocal. 


Fll  ACT  IONS. 


8^} 


EXERCISES, 


D 

vide 

ah 

^>y 

a 

I . 

(I  — 

b 

b 

X 

X 

^+i  by    ^^±i. 


7- 
S. 


ft      a      c  ,     m   ,   n 

-  +  ^  +      by  -  +      -h 
X      II       z         X       y       z 


2. 

4. 
6. 


a;  +  1 


8-  ''y  ¥ 


rt" 


r?       7H  ,      b        II 
.  +~  by 


a 
a  —  b 


b       .  b      ^ 


a 


a  -{-  b         a  —  b       a  -\-  b 


KL'ciprocal    llelatioiis    of  ^lultiplicatioii    aiul 

Division. 

113.  The  fundamental  prinei]ile>s  ol'  tlic  operations  upon 
I'niotions  are  included  in  the  following  summary,  the  under- 
standing^ of  which  will  afford  the  student  a  test  of  his  grasp  of 
till,'  sul)ject. 

1.  The  reci2:)rocal  of  the  reciprocal  of  a  number  is 
('([ual  to  the  number  itself.     In  the  language  of  Algebra, 


1 
a 


—  a. 


2.  The  reciprocal  of  a  monomial  may  be  expresscMl 
by  clianging  the  algebraic  sign  of  its  exponent. 

8.  To  multiply  by  a  number  is  equivalent  to  dividing 
by  its  reciprocal,  and  'vicc  versa.     That  is, 

JV 
N  X  a    or    -r-  =  aK 
i  ' 


and  rice  versa, 


a 

j\  X  -  =  —  • 
a        a 


M 


'•    { 


iD    :  i 


t 


'    I 


t- 


. ;  '^^ 


'       I 


84 


ALGEBRAIC   OPERATIONS. 


4.  When  the  numerator  or  denominator  of  a  fraction 
is  a  proddct  c^'  several  factors,  any  of  these  factors  may 
be  ti-aiisferred  from  one.*  term  of  the  fraction  to  the  other 
by  changing  it  to  its  reciprocal.     That  is, 

1 


air 
'pqr 


he 


p 


ahc 


1 

a 


Or, 


jjqr 
be 


qr 


etc. 


etc. 


abc  __      be      p~^abc 

2)qr  ~  (r^pqr  ~     qr    ' 

5.  MiiUlpUcation  by  a  factor 

greater  than  unity  increases, 
less  than  unity  diminishes. 
Division  by  a  divisor 

greater  than  unity  diminishes, 
less  than  unity  increases. 

6.  («)  When  a  factor  becomes  zero,  the  product  also 
becomes  zero. 

1/3)  AVhen  a  denominator  becomes  zero,  the  quotient 
becomes  infinite.     That  is, 

0  X  rt  —  «  X  0  ==  0. 


a 
0 


infinity. 


Note.  The  following  way  of  expressing  what  is  meant  l)y 
this  last  statement  is  less  simple,  but  is  logically  more  correct: 

If  a  fraction  has  a  fixed  numerator,  no  matter  how 
small,  we  can  make  the  denominator  so  much  smaller 
that  the  fraction  shall  be  greater  than  any  quantity  we 
choose  to  assign. 

EXERCISE. 

If  tlie  numerator  of  a  fraction  is  2,  how  small  must  the 
denominator  he  in  order  that  the  fraction  may  exceed  one 
thousand?  That  it  may  exceed  one  million  ?  That  it  may 
exceed  one  thousand  millions? 


I 


^1*^ 


Ciiii 


action 

s  may 
other 


ict  also 
lotieiit 


emit  by 
correct : 

er  how 
^iiialler 
tity  we 


lUst  tlie 
eed  one 
t  it  may 


BOOK    III. 
OF    EQUA  TIONS. 


CHAPTER    I. 
THE     REDUCTION     OF    EQUATIONS. 


it>. 


Definitions. 

')ef.     An  Equation  is  a  statenK^nt,  in  the  han- 
guage  of  Algebra,  that  two  expressions  are  equjil. 

111:.  Def.  The  two  equal  expressions  are  called 
Members  of  the  equation. 

115.  Def.  An  Identical  Equation  is  one  which  is 
tiue  for  all  values  of  the  algebraic  symbols  wliicli  enter 
into  it,  or  which  has  numbers  only  for  its  members. 

Examples.    The  equations 

14  4.  9  -  29  —  G, 
5  +  13  _  3  X  4  -  G  =  0, 
which  contain  no  algebraic  symbols,  are  identical  equations. 
So  also  are  the  equations 

X  =  X, 

X  —  X  =  0. 
{x  4-  a)  {x  —  (()  —  x^  —  (fi, 
(1-f  y)(l-y)-l+.V^  =  0, 

because  they  are  necessarily  true,  whatever  values  "we  assign  to 
.r,  a,  and  y. 

Kem.  All  the  equations  used  in  the  preceding  two  books 
to  express  the  relations  of  algebraic  quantities  are  identical 
ones,  because  they  are  true  for  all  values  of  these  (pumtitics. 


I 


1       I',. 


.f  ■  ' 


il  ^ 


80 


EQUATIONS. 


*.. 


I,     :j^  , 


^     :n 


110.  Dcf.  An  Equation  of  Condition  is  one  wliicli 
can  1)(^  true  only  when  the  algebraic  symbols  are  equal 
to  (Certain  quiuitities,  or  have  certahi  relations  among 
themselves. 

Examples.     The  cijiiulion 

x-\-i)  =  22 

can  be  true  only  when  x  is  equal  to  10,  and  is  therefore  an 

equation  of  condition. 

The  e(iUiitioii 

X  -i-  b  =  a 

can  be  true  only  when  x  is  equal  to  the  difference  of  the  two 
quantities  a  and  b. 

Rem.  In  an  equation  of  condition,  some  of  the  quantities 
may  be  supposed  to  be  known  and  others  to  be  unknown. 

117.  Drf.  To  Solve  an  equation  means  to  lincl 
some  number  or  algebraic  expression  which,  being  sub- 
stituted for  the  unknown  quantity,  will  render  the 
ec  nation  identically  true. 

.'his  value  of  the  unknown  quantity  is  called  a  Root 
Oi  the  equation. 

EXAMPLES. 

1.  The  number  3  is  a  root  of  the  equation 

2.6'2  _  18  z=  0, 

because  when  we  put  3  in  place  of  x,  the  equation  is  satisfied 
identically. 

2.  The  expression is  a  root  of  the  equation 

2cx  —  ■i:a  +  zh  =  0, 

when  X  is  the  unknown  quantity,  because  when  we  substitute 
this  expression  in  place  of  .t,  avc  have 

0, 


2c ' 


or  4fl!  —  26  —  4rt  4-  2b 

which  is  identically  true. 


AXIOMS. 


87 


RvM.  It  is  common  in  Elenu'iitiiry  Al<]^cl)rii  to  r{>))ro.S('iit 
unknown  qiuintitict-i  l)y  tlio  lust  letters  of  tlio  iilj)lialu't,  und 
(liumtitii'S  siipixjsed  to  he  known  hy  tlio  first  let  tors.  \h\t  this 
is  not  at  Jill  necessary,  and  the  student  should  accustom  him- 
self to  regard  any  one  symhol  as  an  unknown  quantity. 

Axioms. 

118.  Def.  An  Axiom  is  a  proposition  which  is 
taken  for  granted,  without  proof. 

Equations  are  solved  hy  o})erati()ns  founded  npon  the  fol- 
lowing axioms,  which  arc  self-evident,  and  so  need  no  proof. 

Ax.  I.  If  equal  quantities  be  added  to  the  two 
members  of  an  equation,  the  members  will  still  be  equal. 

Ax.  II.  If  equal  quantities  be  subtracted  from  the 
two  members  of  an  equation,  they  will  still  be  equal. 

Ax.  III.  If  the  two  members  be  multiplied  by  equal 
factors,  they  will  still  be  equal. 

Ax.  IV.  If  the  two  members  be  divided  by  equal 
divisors  (the  divisors  being  different  from  zero),  tiny 
will  still  be  equal. 

Ax.  V.     Similar  roots  of  the  two  members  are  equal. 

These  axioms  may  he  summed  up  in  the  single  one, 

SiniiUn^  opcroMons  upon  eq^ial  quantities  ^ivo  equal 
Tcsults. 

119.  An  algebraic  equation  is  solved  by  performing 
such  similar  operations  upon  its  tw^o  members  that  the 
unknown  quantity  shall  finally  stand  alone  as  one 
member  of  an  equation. 

Operations  of  Addition  and  Subtraction— Trans- 
posing Terms. 

130.  Theorem.  Any  term  may  be  transposed  from 
one  member  of  an  equation  to  the  other  member,  if  Its 
sign  be  changed. 


•I 


M' 


t 
I 


88 


h'QL/ATlOX.S. 


Proof.     Let  us  ])ui,  it)  uccoroliince  will  g  41,  Jid  IVin., 
/,  any  term  of  eitluT  nic'inl)er  ol'  the  L(|iiatiuu. 
rt,  all  tlio  other  terms  of  tlie  bamo  member. 
b,  tlie  opposite  member. 
The  ecpuitiou  is  then 

a-\-  t  =  h. 

Now  subtract  /  from  Itoth  skies  (Axiom  II), 

a  ^  t  —  I  :=  h  —  I ',' 
or  by  reduction,  a  -—  h  —  I. 

Tiiis  e(|uation  is  the  same  as  the  one  from  wliicli  wc  started, 
cxcejjt  tliat  /  has  been  transposed  to  the  second  member,  Avith 
its  sign  changed  from  4-  to  — . 

If  the  e(puition  is 

h  —  t  =.  n, 

wc  may  add  t  to  both  members,  wliich  would  give 

h  =  a  -\-  t. 

NUMERICAL      EXAMPLE. 

The  learner  will  test  eacli  side  of  the  Iblluwing  equations  : 

19  +  3-0  +  -1  =:  7  +  10. 
194_.3_0  -  7+10-4. 
19  +  3  —  7  +  10-4  +  0. 

3  =  7-1-10-4  +  9-19. 
0  =  7  +  10—4  +  9-19-3. 

All  the  terms  of  either  member  of  an 
equation  may  be  transposed  to  the  other  member, 
leaving  only  0  on  one  side. 

Example.    If  in  the  equation 

h  =  a-\-  t, 
we  transpose  h,  we  have    0  =  «  +  ^  —  J. 

By  transposing  a  and  t,  we  have 

J)  —  a  —  t  =z  0. 

133.  Clianging  Signs  of  Members.  If  we  change  the  signs 
of  all  the  terms  in  both  members  of  an  equation,  it  will  still 
be  true.    The  result  will  be  the  same  as  multiplying  both 


Trans[)osing  4, 

«  19, 

3, 

131.  Rem. 


UIIDVCTION. 


89 


iiu!iiibers  by  —  1,  or  tninsposinj^  all  tlio  Utiiis  ol'  each  member 
to  the  other  side,  iiiul  then  exeluing.ag  the  terms. 

ExAMPi.K.     Tlie  efiiuition 

J ;  +  8  =  1 1  +  1 4 

may  1^'  transformed  into     0  =  ll-fl-i  —  17  —  8, 

or,  0~  —11  —  14  +  17-1-8, 

or,      -17  — 8  =  -  11  -  14. 


. 


Oper.ation  of  3Iiiltiplieation. 

1*33.  Clear i n(j  of  FradioHK.  'I'ho  operation  of  multii)li- 
ciition  is  usually  performed  upon  the  two  sides  of  an  e(puition, 
ill  order  to  elear  the  ('(piation  of  I'raetions. 

To  clear  an  equation  of  fractions: 

First  Method.  Multlphj  its  nivnihcrs  hij  the  least 
roninwJb  multiple  of  all  Us:  (loioDuiiafoj's. 

Second  Method.  Maltiphj  its  inemhci  -  hy  each  of 
the  (leiioniiiiatnrs  in  succession. 

Rem.  1.  Sometimes  the  one  and  sometimes  the  other  of 
these  methods  is  the  more  convenient. 

Rr.M.  'I.  Tlio  operation  of  clearinf]^  of  fractions  is  similar 
to  tluit  of  reducing  fractions  to  a  common  denominator. 

Example  of  First  Method.  Clear  from  fractions  the 
eqnatioix 

XXX 

4  +  0  +  8  =  *"• 

Here  24  is  the  least  common  multiple  of  the  denominators. 
Multiplying  each  term  by  it,  we  have, 

e.-r  +  4.r  +  3;c  —  G24, 
or  13a;  zn  624. 

Example  of  Second  Method.     Clear  the  equation 

a  a  c 


X 


a 


+  '   =  0. 

X  +  a      X 


Multiplying  by  x  —  a,  we  find 


ax  —  «2      (^x  —  ea       „ 
a  ■[ 1 =  0. 


i: 


■\  H 


n 


X  -{•  a 


X 


1)0  h:qVATU)N8. 

Mul(ii»lyiiif?  l)y  x  -f  a, 


(IX  +  (I-  -j-  ((.r.  —  a^  -\ - 

X 

\ivd[iv'u\<i;  antl  iniiltijdyiui,'  l)y  x, 

"iax^  +  cx^  —  cit^  =  0. 

EXERCISES. 

Clear  tlic  following  cfiiuitioiis  of  fniofioiifs 


=  0. 


—  G  =  0. 


x 


—  -n 


2 


+ 


=  0. 


// 


ah 


+  •      4-    ,    =: 


(( 


(m 


X 


x 


X  —  a      X  ■\-  a 
X  -\-  a       x'^  -i-  2ax 


1. 


a 


a 
x 


% 


X  —  (I 


X  —  a 


a 


II.        —  ■    = 


10. 


12. 


X  —  (i  X  -[•  b 

X  —  'i  _x  +  'i 

X  —  5  X  +  b 

X  —  a  X  -j-  (I 


y 


X  -\-  a 


a 


+ 


a 


0. 


13- 


X 


+ 


// 


a  —  b       I)  —  a 


—  =  z. 


Here  tlie  second  term  Ih  the  same  as 


-y 

a  —  b 


14. 


X  +  a 


a 


a 


Iletluetioii  to  the  XoriiuU  Form. 


12t.  Dff.    An  equation   is  in  its  Normal  Form 

wlion  its  terms  are  reduced  and  arranged  according  to 
the  2)owers  of  the  unknown  quantity. 

In  tlie  normal  form  one  member  of  the  e(|mition  is  expressed 
as  an  entire  funetion  of  the  unknoAvn  quantity,  and  the  other 
is  zero.     (Compare  §§  50,  70.) 

To  reduce  an  equation  to  the  normal  form : 

I.  Transpose  all  the  tcnns  to  one  nvemher  of  the  equa- 
tion, so  as  to  leave  0  as  the  other  meviber. 


liKDUCTlON. 


01 


=  0. 


II.  CI  rat'  the  cqiKtHoii,  af  fntctinnfi. 

III.  i'lrar  flic  njndthni  of  ftftrcnUusrs  hij  perform i mi 
(ill  llir  opi-nifions  inilirn, 

IV.  ('ollrcf  ((irii.  net  of  firms  ronfdiniiifj  like  /jowrrs 
oj'  till'  itiiknoirii  i/fiKufifi/  info  a  sinjjlr  our. 

\ .  Diriilr  hij  an  if  ronnnoii  fuvlor  wiiirli  dors  not  ron- 
hiiii  flir  iinK'iioirn,  (/iKtnfifi/. 

llVM.  Tliis  order  of  o[)orjiti()ii.s  nuiy  hv  deviiitt'd  fV(»ni 
:i((nrdin«f  tocirciiiiistant'cs.  Afteru  littk'  prjictice,  tlie  HtiuU'iit 
iii;iy  take  tho  shortest  way  of  reaching  the  result,  without  re- 
tfpcet  to  rules. 

EXAMPLES. 

I.  lieduce  to  tiio  normal  Torin 

(x  -  2)  (./•  -  li)  ^  {x  +  2)  (a:  +  4)_ 

X  —  0  X  +  ij 

X    Clearing  of  fractions, 

(x  +  5)  {x  —  2){x  —  3)  =  {x  —  5)  {x  f  3)  (x  +  4). 
)l.  Performing  the  imlicated  operations, 

0^  _  lo.f  4-  ;}()  =:  jf^  +  .t2  —  nx  —  40. 

3.  Transposing  all  the  terms  to  tho  second  member  and 

reducing, 

0  =z  X?  —  ^x  —  70, 

Avliicli  is  tho  normal  form  of  tho  equation. 

Rem.  Had  we  transposed  the  terms  of  the  second  member 
to  the  first  one,  the  result  would  have  been 

_  a;2  _^  3^  +  70  =  0. 

Either  form  of  the  equation  is  correct,  but,  for  the  sake  of 
uniformity,  it  is  customary  to  transpose  the  teiais  so  that  tho 
coetJicient  of  tho  highest  power  of  x  shall  be  positive.  Ii  it 
comes  out  negative,  it  is  only  necessary  to  change  the  signs  of 
all  the  terms  of  the  equation. 

Ex.  2.     Reduce  to  the  normal  form, 

bmx'^         2nx  Smx^ 

X  —  a      x  +  a      x^  —  cv^ 


%mx  —  5rt. 


•i 


'  ' '  . 

»    »  !1 


"         'i      \ 


92 


EQUATIONS. 


\n 


1.  Transposing  to  the  first  member, 

bmx^         2ax  'Sinx^ 

—      ,       ,,  — 7i  —  2mx  +  oa  =  0. 

X  —  a      X  -\-  a      x^  —  a^ 

2.  To  clear  of  fi-actions,  we  notice  that  the  least  common 
mnltiplc  of  the  denominators  is  x^  —  tfi.  Multiplying  each 
terra  by  this  factor,  we  have, 

bmx^{x+a)—Zax{x—a)—^mx^—)lmx{x'^—(e)^ba{x^—a^)  =  0. 

3.  Performing  the  indicated  ojierations, 

5)ux^ + bamx^ — 2ax^  +  2a^x —3mx^—2 mx^  +  2ahnx  y bax^ — 5a^=z0. 

4.  Collecting  like  powers  of  x,  as  in  §  7G, 

(3ff  +  6am)  x^  +  {2a^  +  2a^m)  x  —  5a^  =  0. 

5.  Every  term  of  the  equation  contains  the  factor  a.  By 
Axiom  IV,  §  118,  if  both  members  of  the  equation  be' divided 
by  a,  the  equation  will  still  be  true.  The  second  member 
])eing  zero,  will  remain  zero  when  divided  by  a.  Dividing 
both  members,  we  have 

(3  +  5m)  x^  +  2a  (1  +  m)  x  —  6a^  =  0, 

which  is  the  normal  form. 


EXERCISES. 


i.'»i 


Reduce  the  following  equations  to  the  normal  form,  x,  y, 
or  z  being  the  unknown  quantity  : 


I. 

3- 
4. 

5- 
6. 

7- 


'df±2y  _  y-J 

7      ■  ~      2     * 
x-H    _  2x  +  6 
2a;  +  10  "~  4a;  — 2" 
of  _  'Sa^x  +  2«3 


2. 


X  —  a       X  -\-  a 

■  ■  —  ■       '"    — -  '■  ■ 

X  -\-  a  X 


2x  -\-  a 


x^  —  bax  = 


7a;^  —  bax^ 
2x  —  a 


a-y      a  +  y      ay- 


a  ■\-'b      h  -Y  z       a  ^  z 


0. 


+ 


aH 


a  —z      a^  —  x^ 


z^  —  a^ 


REDUCTION. 


93 


7  +  -  +  -V,  +  -3  =  0. 


a 


+ 


a' 


+ 


«• 


X  —  a      x^  —  aJ^      x^  —  a^ 


=  1. 


10. 


II. 


13- 


14. 


15- 


+ 


b^ 


+ 


b* 


¥ 


c  —  z      <?  —  'i?      d^  —  '^       &  —  z^ 


16. 


a              b 

,       1       X  —  a 
0 

X 

12. 

a               a^ 

a a^ -„ 

X                 x^ 

«3 
0^ 

3z             5z^ 

1 

'^'2       ''-I 

~  Z 

ax 

bx 

1    i     1 

X  -\-  a 

1 

1 

'   X  —  a 

a          b 
X      a  —  X 

a 

m 


M 


n 
nx 

X 


x  +  - 

X 


b 

X 


a 


.  I 


'    I 


X 


Degree  of  Equations. 

125.  Def.  An  equation  is  said  to  be  of  the  nJ^^  de- 
gree when  n  is  the  highest  power  of  the  unknown 
quantity  which  appears  in  the  equation  after  it  is  re- 
duced to  the  normal  form. 


i 


I 


EXAMPLES. 

The  equation       Ax  -{■  B  =z  0      is  of  the  first  degree. 

Ax^  ^  B  =  (i      "       "  second    " 

J^  ji^  Bx+  C  =0      "       "  third       " 
etc.  etc. 

An  equation  of  the  second  degree  is  also  called  a 
Quadratic  Equation. 


ij^ 


94 


EQUATIONS  OF  THE  FIRST  DEGREE. 


An  oqnation  of  the  third  degree  is  also  called  a 
Cubic  Equation. 

Example.     The  o(iiiation 

ax^  -\-  bx^y^  ■-  y^  +  «^z  =  0 
is  a  qnadra^ic  eqiuition  in  x,  because  x/^  is  of  the  higliest  power 
of  ^  which  enters  into  it. 

It  is  a  cubic  equation  in  y. 

It  is  of  the  first  degree  in  z. 


-^-•-t- 


CHAPTER    II. 

EQUATIONS    OF    THE    FIRST    DEGREE    WITH    ONE 
UNKNOWN    QUANTITY. 

136.  Remark.  By  the  preceding  definition  of  the  degree 
of  an  equation,  it  will  be  seen  that  an  equation  of  the  first 
degree,  Avith  x  as  the  quantity  supposed  to  be  unknown,  is  one 
which  can  be  reduced  co  the  form 

Ax  -\-  B  =  0,  (a) 

A  and  B  being  any  numbers  or  algebraic  expressions  whicli 
do  not  contain  x. 

Such  an  equation  is  frequently  called  a  Simple  Equ.'iciuii. 

Solution  of  Equations  of  the  First  Dc',5ree. 

137.  If,  in  the  above  equation,  we  transpose  tbo  term  B 
to  the  second  member,  we  have 

Ax  —  —B. 

If  we  divide  both  members  by  A  (§  118,  Ax.  IV),  wv  have, 

B 
A 


X  = 


Here  we  have  attained  our  object  of  so  transforming  the 
equation  that  one  member  shall  consist  of  x  alone,  and  the 
other  member  shall  not  contain  x. 


ONE    UNKNOWN   QUANTITT. 


95 


JJ  . 


To  prove  that is  the  required  value  of  x,  we  substi- 
tute it  for  X  ill  Uie  equation  {a).     The  equation  then  becomes, 


or,  by  reducing, 


-  B  +  B  =  Q, 


B 


an   ei|uation   which  is    identically  true.     Therefore, 7  is 

the  required  root  of  the  equation  {(i).     (§  117,  Dvf^ 

138,  In  an  ecj nation  of  the  first  degree,  it  will  be  unneces- 
sary to  reduce  the  equation  entirely  to  the  normal  form  by 
transposing  all  the  terms  to  one  member.  It  will  generally  l)e 
more  convenient  to  place  the  terms  which  do  not  contain  x  in 
the  opposite  member  from  those  which  arc  multiplied  by  it. 


Example.    Let  the  equation  be 

mx  -\-  a  ^  nx  4-  h. 


(1) 


"We  may  begin  by  transposing  a  to  the  second  member  and 
nx  to  the  first,  giving  at  once, 


or 


mx  —  nx  ■=.  h  —  a, 
{m  —  n)  X  =  Z>  —  a, 


without  reducing  to  the  normal  form.  The  final  result  is  the 
same,  whatever  course  we  adopt,  and  the  division  of  both 
members  by  m  —  n  gives 


X  = 


h-a 
m  —  n 


131).  The  rule  which  may  be  followed  in  solving  equations 
of  the  first  degree  with  one  unknown  quantity  is  this: 

I.  Clear  the  equation  of  fractious. 

II.  Transpose  the  terms  which  are  imdtipliecl  hy  the 
unhnoivn  quantity  to  one  member ;  those  which  do  not 
contain  it  to  the  other. 

III.  Divide  hy  the  total  coefficient  of  the  unknown 
quantity. 


i; 


4' 

si 


'i^r 


■f  '  \ 


I  '  ^1 
'    if 


1  ,'i 


96 


EQUATIONS   OF   THE  FIRST  DEGREE. 


Note.  Rules  iu  Algebra  are  given  ouly  to  eiiublc  the  beginner  to  go 
to  work  in  a  way  which  will  always  be  sure,  though  it  may  not  always 
be  the  shortest.  In  solving  equations,  he  should  emancipate  himself 
from  the  riiUis  as  soon  as  possible,  and  be  prepared  to  solve  each  equa- 
tion presented  by  such  jjrocess  as  appears  most  concise  and  elegant.  No 
operation  upon  the  two  members  iu  accordance  with  the  axioms  ^§  118) 
can  lead  to  incorrect  results  (provided  that  no  quantity  which  becomes 
zero  is  used  as  a  multiplier  or  divisor),  and  the  student  is  therefore  free 
to  operate  at  his  own  pleasure  on  every  equation  presented. 


I.  Given 


EXAMPLES, 

ax 


bij 


=  1. 


It  is  required  to  find  tlie  value  of  each  of  tlie  quantities  a, 
h,  X,  and  y,  in  terms  of  the  others. 
Clearing  of  fractions,  we  have 

ax  =1  ly. 

To  find  a,  we  divide  by  x,  which  gives 

dy 

X 

To  find  hf  we  divide  by  y,  which  gives 

ax 

—  =  0. 

y 

To  find  x,  we  divide  by  a,  which  gives 

by 
a 

To  find  y^  we  divide  by  h,  which  gives 

ax 

T  =  2'- 

Thus,  when  any  three  of  the  four  quantities  a,  b,  x,  and  y, 
are  given,  the  fourth  can  be  found. 

2,  Let  us  take  the  equation, 

x-7    _  2a;  +  6 
2a;  +  10  ~  4:x  —  2 

Clearing  of  fractions,  we  have 

4?;2  _  30x  +  U  =  ^x^  +  33a;  +  60. 


ONE   UNKNOWN  QUANTITY 


97 


Transposing  und  reducing, 

—  Q'2x  =  46. 
Dividing  both  members  by  —  02, 

^  _  _  4(3 

62  ~  ~  62 


X  = 


33 


"1 


This  result  should  now  bo  proved  by  computing  the  value  of  both 


members  of  the  original  equation  when 

X        X        ax        1 

"^      m.       n 


23  . 


31 


is  substituted  for  x. 


b       m 

Proceeding  in   the  regular  way,  we  clear  of  fractions  by 
multiplying  by  vinh.     This  gives 

nbx  -\~  mbx  =  amnx  —  nb. 

Transposing  and  reducing, 

{nb  +  mb  —  amii)  x  ■=  —  nb. 

Dividing  by  the  coefficient  of  x, 

nb  _  7ib 

nb  +  mb  —  amn       amn  —  ynb  —  nb 

These  two  values  are  equivalent  forms  (§  100). 

But  we  can  obtain  a  solution  without  clearing  of  fractions. 

Transposing  -r- ,  we  have 

X       X      ax  1 

m       n        b   ~       7u' 

which  may  be  expressed  in  the  form 

n       1       a\  1 

(-  H y)^  = 

\tn      n      bl  m 

Dividing  by  the  coefficient  of  x, 

1 

m 


x  = 


1^      1 

m       n 


a 
b 


This  expression  can  be  red.  ced  to  the  other  by  §  110. 
7 


I         ! 


iij 


'■■( 


I.'!  :| 


!■' 


.r\    i] 
4< 


n  ■■! 


98 


EQUATIONS   OP   THE  FIRST  DEGREE. 


EXERCISES. 

Find  the  values  of  x,  y,  or  u  in  the  following  equations : 
o  —  3.^        8,/-  —  !) 


I. 

3- 

5- 

7- 

9- 
1 1. 

12. 

13- 
14. 


18. 

19. 

20. 


2 


3 


X         X  X 

1  +  a  +  3  =  ^«- 

■"  +  ■!-■"  =  1. 

C^  6*  C 

i«         ?«         if. 
o         4        0 


2.     —  X  =  a 
X  -\~  ^  f ) 


4- 
6. 


x  —  1 
?^  —  5 


=  9. 
-  15. 


1       1 

-  J 


8.     rt  —  kv  =  h  +  nx. 

10.       ^X  H -—  z=z  X, 

o 


_a _      c 

c  —  X  ~  a  —  x' 

X  —  1  _a;  — 2 
X ~2~ x~3 

-y  -  a  —  b. 

1  1 


r?;  — 5 
^  — G 

1 


x  —  6 
1 


.r  —  2       a;  —  4       a;  —  6       a-'  —  s' 


■^-     ^(-l)-3(-^-4)+i(^-I)  =  0. 


16. f- =: 

a       b  —  a       b  ^  a 

X      1 
17.     ax  -\-  b  :=!  -  -v-  -. 
a       b 


II  ~  a      u  —  b      u 

-| —   -      _j — 


~  _  u  —  {a  J^b  ^  c) 
abc 


m  +  n. 


h  c  a 

m  (x  +  a)       n  {x  +  b)  _ 
X  -\-b      "^     X  -{-  a~ 

{.v-ay  -^-{x-br  +  {x-cY  =.  3  (rr-a)  (.',_^)  ^x-c). 

Find  the  values  of  each  of  the  four  quantities,  a,  b,  c,  and 
d,  111  terms  of  the  other  three,  from  the  equations 

d  -  ah 


21. 


a 


b  -  c 


+  r^-^  =  0.        22.     ^^J+1=.0. 


>■! 


ONE    UNKNOWN   QUANTITY. 


Frobleins  leading-  to  Simple  Eciuations. 


1)9 


i;>().  Tlie  first  difficulty  which  the  beginner  moots  witli  in 
(lie  solution  of  an  algebraic  problem  is  to  state  it  in  the  form 
of  an  equation.  This  is  a  process  in  which  tlie  student  must 
(Ie})end  ui)on  his  own  powers.  Tlie  following  is  the  general 
phiti  of  proceeding  : 

1.  Study  the  ])roblem,  to  ascertain  what  rinantities  in  it 
are  unknown.  There  may  be  several  such  quantities,  bnt  the 
])roblems  of  the  present  chapter  are  such  that  all  these  (|uan- 
tities  can  be  expressed  in  terms  of  some  one  of  them.  Select 
that  one  by  which  this  can  be  most  easily  done  as  the  unknown 
([uantity. 

'I.  Represent  this  unknown  quantity  by  any  algebraic  sym- 
l)ul  whatever. 

It  is  common  to  select  one  of  the  last  letters  of  the  alpha- 
bet for  the  symbol,  but  the  student  should  accustom  himself 
to  work  ec[ually  well  with  any  symbol. 

3.  Perform  on  and  with  these  symbols  the  operations  re- 
quired b)  the  iiroblem.  These  operations  are  the  same  that 
-would  be  necessary  to  verify  the  adopted  value  of  the  unknown 
((uantity. 

4.  Express  the  conditions  stated  or  implied  in  the  problem 


In'  means  of 


f 


The 


equation, 
solution  of  this  equation  by  the  methods  already 


explained  will  give  the  value  of  the  unknown  quantity.  It  is 
always  best  to  verify  the  value  found  for  the  unknoAvn  (quan- 
tity by  operating  upon  it  as  described  in  the  equation. 


'.  1 


[x—c). 
c,  and 


EXAMPLES, 

I.  A  sum  of  440  dollars  is  to  be  divided  among  three  people 
so  that  the  share  of  the  second  shall  be  30  dollars  more  than 
tliat  of  the  first,  and  the  share  of  the  third  80  dollars  less  than 
those  of  the  first  and  second  together.  What  is  the  share  of 
each  ? 

Solution.  1,  Hero  there  are  really  three  unknown  quantities,  hut 
it  is  only  necessary  to  represent  the  share  of  the  first  by  an  unknown 
evinbol. 


iiXm',mmm,Mi.'iu,. 


100 


EQUATIONS   OF   THE   FlRHT  DEGREE. 


i- 


2    Tlieref(iru  let  us  put 

X  ■=.  share  of  the  first. 

8.  Then,  by  the  terms  of  the  statement,  tlie  share  of  the  second  will  be 

X  ^-  30. 

To  find  the  share  of  the  thu  \  we  add  these  two  together,  which  makes 

1x  +  30. 
Subtracting  80,  we  have 

'Xx  —  50 
as  the  share  of  the  third. 

We  now  add  the  three  shares  together,  thus, 

Sharo  of  first,  x 

«       "  second,      a;  +  30 
"       "  third,       "Ix  —  50 

Shares  of  all,  ^x  —  JiO 

4.  By  the  conditions  of  the  problem,  these  three  shares  must  together 
make  up  440  dollars.  Expressing  this  in  the  form  of  an  equation,  we 
have 

4^•  —  20  =  440. 

5.  Solving,  we  find 

a:  =  115  =  share  of  first. 

Whence,  115  +  30  =  145  =  share  of  second. 

115  +  115  _  80  i:=  ISO  =  share  of  third. 

Sum  =  440.     Proof. 

Ex.  2.  Divide  the  number  90  into  four  parts,  snch  that 
the  first  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  shall  all  be  equal 
to  the  same  quantity. 

Here  there  are  really  five  unknown  quantities,  namely,  the  four  parts 
and  the  quantity  to  which  they  are  all  to  be  equal  when  the  operation  of 
adding  to,  subtracting,  etc.,  is  performed  upon  them.  It  will  be  moi^t 
convenient  to  take  this  last  as  the  unknown  quantity.  Let  us  therefore 
put  it  (Hjual  to  u.     Then, 

Since  the  first  part  increased  by  3  must  be  equal  to  w,  its  value  will 

be  u  —  2. 

Since  the  second  part  diminished  by  2  must  be  equal  to  ti,  its  value 

will  be  u  +  2. 

11 
Since  the  third  part  multiplied  by  2  must  be  w,  its  value  will  be  ^  • 

Since  the  fourth  part  divided  by  2  must  make  u,  its  value  will  be  2m, 


ONi:    UXhWOWN   QUANTITY. 


101 


Adding  these  four  [uirta  up,  tlicir  sum  is  found  to  \w 


0'/ 


\\y  the  conditions  of  the  problem,  this  sum  must  make  up  the  nam- 
]mi-  UU.     Therefore  we  huvo 


9w 
2 


=  90. 


Solving  this  equation,  we  find 

It  =  20. 
Therefore 

1st  part  =  u  —  2  =  18. 

2d  "  =  K  -{-2  =  22. 
3d  "  =  u-^2  =  10 
4th    "     —  2u        -  40. 

The  sura  of  the  four  equals  90  as  required,  and  the  first  part  increased 
by  2,  the  second  diminished  by  3,  etc.,  all  make  the  number  20,  as  re- 
el ui  red. 

PROBLEMS    FOR    EXERCISE. 

1.  What  number  is  tliat  from  whicli  avo  ohtain  the  same 
result  Avhelher  we  multiply  it  by  4  or  subtract  it  from  100? 

2.  What  number  is  that  which  gives  the  same  result  when 
^\Q  divide  it  by  8  as  when  we  subtract  it  from  81  ? 

3.  Divide  284  dollars  among  two  people  so  that  the  share 
of  tlie  first  sluiU  be  three  times  that  of  the  second  and  $10 

more. 

4.  Find  a  number  such  that  \  of  it  shall  exceed  \  of  it 
by  12. 

5.  A  sliepherd  descrilies  the  number  of  his  sheep  by  saying 
tliat  if  he  had  10  sheep  more,  and  sold  them  for  5  dollars  each, 
lie  would  have  G  times  as  many  dollars  as  he  now  has  sheep 
lluw  many  sheep  has  he  ? 

6.  An  applewoman  bought  a  number  of  apples,  of  which 
GO  proved  to  ne  rotten.  She  sold  the  remainder  at  the  rate  of 
2  for  3  cents,  and  found  that  they  averaged  her  one  cent  each 
for  tlie  whole.     How  many  had  she  at  first? 

7.  If  you  divide  my  age  10  years  hence  by  my  age  20  years 
ago,  you  will  get  the  same  quotient  as  if  you  should  divide  my 
present  age  by  my  age  26  years  ago.     What  is  my  present  age  ? 

8.  Divide  1500  among  A,  B,  and  C,  so  that  B  shall  have 
$20  less  than  A,  and  C  $20  more  than  A  and  B  together. 


'I    :  j    ,J 

.J 


ii 


n^^ 


102 


EQUATIONS   OF    T1I/-J   FIRST  UKGllEU 


'if 


9.  A  fiitluT  left  810()(>()  to  l)L'  (lividtd  amon^'  his  five  cliil- 
dreii,  (liivc'tiii,!,^  llitit  oacli  shoiikl  ivcoive  !?<,">()()  iiu»i-o  than  thf 
next  younger  one.     What  was  the  share  of  each  ? 

10.  A  man  is  0  years  older  tlian  his  wife.  After  tliov  hiivc 
been  married  1^  years,  8  times  lier  ji^'e  would  make  )'  tinu^ 
liis  acfe.     Wiiat  was  their  a^re  when  married? 

11.  or  three  brothers,  the  vounijest  is  S  vers  vonnffer  thiiii 
the  second,  and  the  ehlcst  is  as  ohi  as  tiie  othe  •  two  to;^i'ther. 
In  10  years  the  sum  ol'  tiieir  ages  will  be  l^iO.  What  are  their 
present  ages*'' 

12.  The  head  of  a  fish  is  0  inches  long,  the  tail  is  as  lon:r 
as  the  head  and  hall'  the  body,  and  the  body  is  as  long  as  the 
head  and  tail  together.     What  is  the  whole  length  of  thelishr 

13.  In  dividing  a  year's  profits  between  three  ])artners,  A, 
B,  and  C,  A  got  one-fourth  and  SlOO  more,  B  got  one-thini 
and  '^'M){)  more,  and  C  got  one-lil'th  and  ^^^  more.  What  was 
the  sum  divided  ? 

14.  A  traveller  in([uiring  the  distance  to  a  city,  wiis  told 
that  after  hv  liad  gone  one-third  the  distance  and  one-lhiid 
the  remainii.g  distance,  he  would  still  have  30  miles  more  to 
go.     Wliat  was  the  distance  of  the  city? 

15.  In  making  a  journey,  a  traveller  went  on  the  first  day 
one-lil'th  of  the  distance  and  8  miles  more  ;  on  the  second  day 
he  Avcnt  one-lift h  the  distance  that  remained  and  15  miles 
more;  on  the  third  day  he  went  one-third  the  distance  that 
remained  and  VI  miles  more  ;  on  the  fourth  he  went  35  miles 
and  finished  his  journey.  What  was  the  whole  distance 
travelled  ? 

16.  WHien  two  partners  divided  their  profits,  A  had  twico 
as  much  as  15.  [f  he  paid  R  ^'.W),  he  would  only  have  half  as 
much  again  as  B  had.     What  was  the  share  of  each  ? 

17.  At  noon  a  ship  of  war  sees  an  enemy's  merchant  vessel 
15  miles  away  sailing  at  the  rate  of  0  miles  an  hour.  How  fast 
must  the  ship  of  Avar  sail  in  order  to  get  within  a  mile  of  the 
vessel  by  0  o'clock  ? 

18.  A  train  moves  away  from  a  station  at  the  rate  of  U 
miles  an  hour.  Half  an  bour  afterward  anotber  train  follows 
it,  running  w  miles  an  hour,  llow  long  will  it  take  the  latter 
to  overtake  it  ? 

19.  What  two  numbers  are  they  of  which  the  difference  is 
9,  and  the  difference  of  their  squares  351  ? 

20.  A  man  bought  25  horses  for  12500,  giving  $80  a  piece 


ONE    UNKNOWN  QUANTITY 


\{)\\ 


f(ir  poor  liorsos  and  ^\'M)  cnicli  for  p^ood  ones.     ITow  many  of 
(,;i(li  kind  did  iio  hiiy  ? 

2\.  A  man  is  7^  yt'ars  older  lliiin  lii.  wife.  In  IT)  years  llio 
sums  ol' their  a.ir*-''^  u  ill  be  (liree  tiii'.es  {\\v  present  a^e  ol"  llie 
wiiV.     What  is  tlic  a^a' of'oaeli  y 

22.  IIow  r.ir  can  a  pei'son  utio  lias  S  hours  to  spare  ritie  in 
\{  coaeli  lit  the  rate  of  <i  miles  an  hour,  so  that  he  can  return  at 
the  rate  ol* -l  miles  an  hour  ami  arrive  home  in  time? 

23.  A  workin<Tf  alone  can  do  a  piece  of  work  in  15  days, 
iuid  B  alone  can  perform  it,  in  \'l  days.  In  what  time  can  thiy 
jierl'orm  it  if  both  work  together  ? 

Mktiiod  of  SoiaTiON'.  Tn  (m<>  day  A  ran  do  ,V.  of  Ihc  wlmlc  work 
and  H  can  do  ,',..     Hence,  both  together  can  do  (,ij+  ,V,)  of  it. 

If  both  together  can  do  it  in  x  daya,  '.^'en  they  can  do     (jf  it  in  1  day. 


1       1 
la  "^  15 


Hence, 

is  the  e(i nation  to  be  solved. 

24.  A  cistern  can  bo  fdled  in  VI  minutes  by  two  pipes  wliieh 
niu  into  it.  One  of  them  alone  will  fill  it  in  )iO  minutes.  In 
what  time  would  the  other  one  alone  till  it  ? 

25.  A  cistern  can  be  emi)tie(l  by  three  pii)es.  The  second 
])ipe  runs  twice  us  much  us  the  first,  an<l  the  third  us  much  as 
the  first  and  second  together.  All  three  together  can  ein))ly 
the  cistern  in  one  liour.  In  what  time  would  each  one  sepa- 
rately empty  it  ? 

26.  A  marketwoman  bouglit  apples  at  the  rate  of  5  for  two 
cents,  and  sold  half  of  them  at  2  for  a  cent  and  tiie  other  half 
ut  ;]  for  a  cent.  Iler  profits  were  50  cents.  How  many  did 
.she  buy  ? 

27.  A  grocer  having  50  pounds  of  tea  worth  00  cents  a 
pound,  mixed  Avitli  it  so  much  tea  ut  60  cents  u  pound  that 
the  combined  mixture  was  worth  70  cents.     How  much  did 

he  add  ? 

28.  A  laborer  was  hired  for  40  days,  on  the  condition  that 
every  day  he  worked  he  should  receive;  $1.50,  but  sliould  for- 
feit 50  cents  for  every  day  he  was  idle.  At  the  end  of  the 
time  $52  were  due  him.     How  many  days  was  he  idle  ? 

29.  A  father  left  an  estate  to  his  three  children,  on  the 
condition  that  the  eldest  should  he  paid  -$1200  and  the  second 
-^SOO  for  services  they  had  rendered.  The  renuiinder  was  to  be 
eijually  divided  among  all  three.     Under  this  urrangement, 


"I,   ! 


,  \ 


''     I 


f 


.( 


i: 


>'   '} 


104 


h'QUATlOXS   OF   Till':   FlltsT   Di:(inKK. 


the  yoniifjcsf.  ^ot  onc-fourtli  of  the  estate.     Wluit  was   tho 
amount  divi<h'(l  ? 

30.  A  person  hiivin<(  a  .sum  of  money  to  divide  umon;; 
three  pcoidc  '^x\\(\  llic  lir-l  one-third  aixi  '^'10  nmre,  tlic  second 
oiie-tliii'd  of  wiiat  was  left  and  %<•.'()  ninri',  and  tiie  tliird  one- 
(iiii'd  orwiial,  was  then  Kd't  and  ><•.'(>  more,  whiedi  exliunsleil  ihe 
uinonnt.     How  nnndi  had  they  to  divide? 

31.  One  shepherd  spent  $7'*0  in  sheep,  and  another  <,'ol  tlie 
panie  !inniher  of  shi'cp  for  ^48(»,  paying'  %'l  a  i)iece  less.  What 
priee  did  each  pay? 

32.  A  crew  which  can  ])ull  at  the  rafe  of  !)  miles  an  hour, 
finds  that  it  takes  twice  as  h)n;^^  to  ;^o  up  the  river  as  to  go 
down.     At  what  rate  (hjes  the  river  How  ? 

33.  A  person  who  ))ossesses  ^1:^()00  employs  a  porti(»n  of 
the  money  in  huildin<^  a  house.  Of  the  money  Avhich  I'cmains, 
lie  invests  one-third  at  four  jx-r  cent,  and  the  otlier  two-thirds 
at  live  per  cent.,  anil  ohtains  from  these  two  investments  an 
annual  income  of  ^'.VXl.     Wiiat  was  the  cost  of  the  iiouse  ? 

34.  An  income  tax  is  levied  on  the  condition  that  the  fh'st 
$000  of  every  income  shall  he  untaxed,  the  next  80OOO  shall 
be  taxed  at  two  ])er  cent.,  and  all  incomes  in  excess  of  $l)(i(M» 
shall  he  taxed  three  ])ereent.on  the  excess.  A  person  linds 
that  hy  a  unilorm  tax  oftwo^jjer  cent,  on  all  incomes  he  would 
save  $"-iOO.     What  was  his  inco'  -c  ? 

35.  At  what  time  hotween  IJ  and  4  o'clock  is  the  mhmte- 
hand  5  minutes  ahead  of  the  hour  hand? 

36.  One  vase,  holding;'  d  gallons,  is  full  of  water;  a  second, 
liolding  /;  ^^allons,  is  full  of  brandy.  Find  the  cai)acity  of  a 
di))per  such  that  whether  it  is  tilled  from  the  first  vase  and  tho 
water  removed  replaced  by  bmndy,  or  tilled  from  the  second 
vase  and  the  latter  then  ill  led  with  water,  the  strength  of  tho 
mixture  will  be  the  same. 

37.  Divide  a  number  w  into  four  such  parts  that  the  first 
part  increased  hy  a,  the  second  diminishetl  by  (t,  the  third 
multiplied  by  a,  and  the  fourth  divided  by  a  shall  all  be  eipuil. 

38.  J3ivido  a  dollars  junong  five  brothers,  so  that  each  shall 
have  u  dollars  more  than  the  uext  younger. 

39.  A  courier  starts  out  from  his  .station  riding  8  miles  an 
houi'.  Four  hours  afterwards  he  is  followed  by  another  riding 
10  miles  an  hour.  How  long  will  it  rerpiire  for  the  second  to 
overtake  the  first,  and  what  will  be  the  distance  travelled? 

If  X  be  the  nuiuln'r  of  hours  required,  the  serond  will  liave  travelled 
X  hours  and  tho  first  (,?'  + 4)  hours  when  they  nieu't.  At  this  time  thej 
must  have  travelled  eciual  distances. 


T^et 

iiiiirs  ( 
}itH  iiij 
on  ii((l 
///I'  pot 

Let 
will  ha 
Since  tl 
the  end 
distance 
overtake 


«olv 


Mull 
which  ii 


This 
tilting  f( 
l)roblem 
<'  =  8,  Ji 

liiid 

wh.ich  is 
To  i] 

shall  no^ 

ferent  su 
(1.)  ] 

of  travel] 


ONE    UNKNOWN  QrANTfTV. 


105 


Problem  of  i\w  Couriers. 

Ijot  us  fifoncnili/.L'  the  prcct'din^'  problem  llius  : 

ltd.  ./  cniirirr  shu'ls  out  from  his  shiHoit  riiUn<J  e 
itiilrsitii  lioiw;  h  /tours  luhr.hr  is  j'olloiriil  liij  iiiiol her 
villi ii'J  ft  milrs  an  /tour.  How  /otto'  iril/  //ir  /iit/rr  /if  in 
tirrrt(t/\'ifto'  /Itr  jirst,  (fti(/  ii'/iul  iri//  /ir  l/ir  (/istaiici'  froit) 
tlir  />oiiit  of  (Irfirtrtitrr. 

Lot  lis  ])ut  t  for  the  lime  re<juiiT(l.  Tluii  llie  first  courier 
will  liiive  travelled  (/-f//)  hours,  und  the  second  /  hours. 
Since  the  lirst  travelled  r  miles  an  hour,  his  whole  distunce  at 
the  end  of  /-f/i  hours  will  l)0  (/  +  //)  r.  In  the  same  way,  tlie 
distance  tr.ivelled  by  the  other  will  he  of.  When  the  hitler 
overtakes  the  former,  the  distances  will  be  ecjual ;  hence, 


at  =  /•(/  +  //). 

Solving  this  cfjuation  with  respect  to  /,  we  llnd 

r/t 


t  = 


n  —  (' 


(1) 


c^) 


Multiplying  by  a  gives  us  the  whole  distance  travelled, 
which  is 

Distance  =       — • 
a  —  c 

This  equation  scdves  every  problem  of  this  kind  by  substi- 
tuting for  rt,  c,  and  //  their  values  in  numbers  sujjposed  in  the 
problem.  For  example,  in  Problem  39,  we  supjwsed  (i  =■  10, 
(-  =  8,  Ji  =  4.  Substituting  these  values  in  equation  (^i),  we 
lind 

which  is  the  number  of  hours  required. 

To  illustrate  the  generality  of  an  algebraic  problem,  wo 
shall  now  inquire  what  values  t  shall  have  when  we  make  dif- 
ferent suppositions  I'cspecting  n,  r,  and  //. 

(1.)  Let  us  suppose  <(  =  r,  or  a  —  c  -=  0.  that  is,  the  rates 
of  travelling  equal.    Then  equation  (^)  will  become 

c/i 


t  = 


'     i 


0' 


■  JT-T  . — IK. 


100 


EQUATIONS   OF   THE  FIRST  DEGREE. 


ii« 


M-  : 


fill  expression  for  infinity  (§  112,  G),  showing  tliat  the  one  coiiriur 
Avould  never  overtake  the  other.     This  is  phiin  enough.     But, 

(2.)  Let  us  supjiose  tliat  the  second  courier  does  not  ride 
80  fast  as  the  lirst,  that  is,  a  less  than  c,  and  a  —  c  negati\('. 

Then  tlie  fraction  — —  will  not  be  infinite,  l)ut  will  he  neija- 

a  —  c  ° 

tive,  because  it  has  a  positive  numerator  and  a  negative  denuin- 

iuator.     It  is  plain  that  the  second  courier  would  never  overtake 

the  first  in  this  case  either,  because  the  latter  would  gain  on 

him  all  the  time  ;  yet  the  fiaction  is  not  infinite. 

What  does  this  mean  ? 

It  means  that  the  problem  solved  by  Algebra  is  more  gen- 
eral, that  is,  involves  more  particular  problems  than  wei\' 
iin})lied  in  the  statement.  If  we  count  the  hours  af/er  the 
second  courier  set  out  as  positive,  then  a  negative  time  will 
mean  so  many  hours  before  he  set  out,  and  this  Avill  bring  out 
a  time  when,  according  to  our  idea  of  the  problem,  the  horses 
were  still  in  the  stable. 

The  explanation  of  the  difficulty  is  this.     Suppose  S  to  be 

the  point  from  which  the  couriers  s  arted,  and  AB  the  road 

{lion Of  which  they  travelled  from 

AS 

8  toward  B.     Suppose  also  that        mm^^m^^^^^mmm^^.^^ 

the   first    courier    started    out 

from  S  at  8  o'clociv  and  the  second  at  12  o'clock.  By  the  rule 
of  positive  and  negative  quantities,  distances  towards  A  arc 
negative.  Now,  because  algebraic  quantities  do  not  commence 
at  0,  but  extend  in  both  the  negative  and  positive  directions, 
the  algebraic  problem  does  not  suppose  the  couriers  to  have 
really  commenced  their  journey  at  S,  but  to  have  come  from 
the  direction  of  A,  so  that  the  first  one  passes  S,  without  stop- 
ping, at  8  o'clock,  and  the  second  at  12.  It  is  plain  that  if  tli- 
first  courier  is  travelling  the  faster,  he  must  luive  passed  the 
other  before  reaching  S,  that  is,  the  time  and  distance  arc 
both  negative,  just  as  the  problem  give<  them. 

The  general  principle  here  involved  may  be  expressed  thiv;: 

In  Ahjchi'K.  roads  and  journey.^,  like  (line,  have  no  hegln' 
ninfj  and  no  end. 


B 


ONE    UNKNOWN   QUANTITY. 


107 


(3.)  Let  us  suppose  Ihnt  the  Cv)uricr,s  start  out  at  the  same 
lime  and  ride  with  the  same  speed.  Tlien  li  and  a  —  c  are 
both  zero,  and  tlie  expression  for  /  assumes  the  Ibrm, 


t  = 


0 


Tliis  is  an  expression  •\vhicli  may  luive  one  value  as  well  as 
another,  and  is  therefore  indeterminate.  The  result  is  correct, 
Ijucause  the  couriers  are  always  together,  so  that  all  values  of 
/  are  equally  correct. 

The  equation  (1)  can  be  used  to  solve  the  problem  in  other 
fi)rms.  In  this  equation  are  four  quantities,  (f,  c,  h,  and  /,  and 
wlicn  any  three  of  these  are  given,  the  fourth  can  he  found. 
There  are  therefore  four  problems,  all  of  which  can  be  solved 
IVom  this  equation. 

First  Photilem,  that  already  given,  in  which  th(>  time 
required  for  oue  courier  to  overtake  the  (>thcr  is  the  unknown 
quantity. 

^^ECOXD  Problem.  A  courier  sets  out  from  a  station, 
riding  c  miles  uu  hour.  After  h  liours  another  folloivs 
Jiini  from  the  same  station,  intending  to  overtake  him 
ill  t  hours.    How  fast  must  he  ride  ? 

The  problem  can  be  put  into  the  form  of  an  equation  in 
tlie  same  way  as  before,  and  we  shall  have  the  equation  (I), 
only  a  will  now  be  the  unknown  quantity.  If  we  use  the 
numbers  of  Prob.  39  instead  of  the  letters,  Ave  shall  have,  in- 
stead of  equation  (1),  the  following  : 

IGa  =  8  (10  +  4)  z=  8-20  =  IGO, 

whence  a  =  10. 

If  we  use  letters,  we  find  from  (1), 

c  {f  +  h) 


a  = 


t 


and  the  problem  is  solved  in  either  case. 

Third  Problem.     The  second  courier  can  ride  Just  a 
miles  an  hour,  and  the  first  courier  starts  out  h  hours 


t;    . 


I  ' 


I  1 


t    ; 


108 


EQUATIONS   OF   THE  FIltlST   DEGREE. 


;i';: 


m 


hefore  li'nn.     How  fast  must  the  latter  vide  in  order  that 
the  other  may  take  t  hours  to  overtake  him? 

Here  Cy  the  rate  of  the  first  conricr,  is  the  unknown  quan- 
tity, and  by  solving  equation  (1)^     j  tind 

at 
t  +  h 

FouitTiT  Problem.  The  swiftest  of  two  couriers  can 
ride  a  miles  an  hour,  and  the  slower  c  Tuilcs  an  Jtour. 
How  long  a  start  must  the  latter  have  in  order  that  tlte 
other  Diaij  require  t  hours  to  overtake  him? 

Here,  in  ecjuation  (1),  h  is  the  unknown  quantity.  By 
solving  the  equation  with  respect  to  h,  we  lind, 

^        cd.  —  ct 

h  =  — -, 

wiiieh  solves  the  problem. 


11 


PROBLEMS    OF    CIRCULAR     MOTION. 

40.  Two  men  start  from  the  same  poirt  to  run  repeatedly 
I'ound  a  circle  one  mile  in  circumference.  If  A  runs  7  milt's 
an  hour  and  B  5,  it  is  required  to  know : 

1.  At  what  intervals  of  time  will  A  pass  B  ? 

2.  At  how  many  different  points  on  the  circle  will  they  be 

together? 

We  reason  thus  :  since  A  runs  3  m.'les  an  hour  faster  than  B,  he  <jieta 
away  from  him  at  the  rate  of  2  miles  a;  hour.  When  he  overtakes  him, 
he  will  have  gainnd  up  )n  him  one  cirenmterence,  that  is,  1  mile.  Thin 
will  require  80  minutes,  which  is  therefore  the  required  interval.  In 
this  interval  A  will  have  gone  round  ii^  and  B  2^^  times,  bo  that  they  will 
be  together  at  the  point  opposite  that  where  they  were  together  'M 
minutes  previous.  Hen  le,  they  are  together  at  two  opposite  points  oi 
the  circle. 

41.  What  would  be  the  answer  to  the  preceding  ques- 
lioii  if  A  should  run  S  mi^.es  an  hour,  and  1>  5? 

42.  Two  race-horses  ruii  round  and  I'onnd  a  course,  the 
one  inakin^^  the  circuit  in  30,  the  other  in  35  seconds.  If 
thev  start  out  too;ether,  liow  lono-  before  they  will  l>e 
to<>-etlier  ao-ani  'i 

Note.    In  x  seconds  one  will  make  .' .  circuit  and  the  other  „_. 

43.  If  one  planet  revolves  round  the  sun  in  T  and  the 
other  in  T'  years,  wliat  will  be  the  interval  between  their 
conjunctions^  i 


tity. 


TWO    UJS KNOWN  QUANTITIES. 


109 


r  that 
.  quan- 


rs  can 
h  hour, 
uct  tJiC 


ty.     By 


peutedly 
7  miles 


they  be 

he  Li'etJ! 
kes  him, 
le.  Thin 
rval.  Ill 
they  will 
rether  'SO 
points  of 

^  ques- 

irsc,  tlio 
ids.  H 
will   1)0 

X 

and  the 
ni  theii" 


CHAPTER    ill. 

EQUATIONS   OF   THE    FIRST    DEGREE    WITH    SEVERAL 
UNKNOWN    QUANTITIES. 


Case  I.    Equations  tvith  Two  Unknoivn  Quan- 

titles. 

\ti2,  Def.  An  equation  of  the  first  degree  with  two 
unknown  quantities  is  one  which  admits  of  being  re- 
duced to  the  form 

ax  -{-  bi/  =  c, 

in  which  cc  and  y  are  the  unknown  quantities  and  cr,  &, 
and  c  represent  any  numbers  or  algebraic  equations 
which  do  not  contain  either  of  the  unknown  quantities. 

Brf.  A  set  of  several  equations  containing  the  same 
unknown  quantities  is  called  a  System  of  Simulta- 
neous Equations. 

Solution  of  a  Pair  of  Siiiiiiltaiieoiis  E<iuatioiis 
containing^  Two  Unknown  Quantities. 

133.  To  solve  two  or  more  simultaneous  equations, 
it  is  necessary  to  combine  them  in  such  a  way  as  to 
iorm  an » equation  containing  only  one  unknown  quan- 
tity. 

134.  Def.  The  process  of  combining  equations  so 
that  one  or  more  of  the  unknown  quantities  shall  dis- 
a])pear  is  called  Elimination. 

The  term  "elimination"  is  used  because  the  unknown 
quantities  which  disappear  are  eliminated. 

There  are  three  methods  of  eliminating  an  unknown  quan- 
tity from  two  simultaneous  equations. 


•    I 


-;] 


■■n'-^' 


110 


EQUATIONS   OF   THE   FIRST  DEGREE. 


tl 


'ti' 


i 


Eliiniiuitioii  by  Comparison. 

1*^5.  Rule.  Solve  each  of  the  equations  with  respect 
to  one  of  the  unhnouni  quantities  and  put  the  two  values 
of  tlie  wnknown  quantitij  thus  ohtaineil  equal  to  each 
other. 

This  will  give  an  equation  with  onlij  one  unhnou-ii 
quantitij.  of  ivhicJi  the  value  can  he  found  from  the 
equation. 

TJie  value  of  the  other  unknown  quantitij  is  then 
found  hy  substitution. 


Example.     Let  tlie  equations  be 

ax  ■\-  ly  =1  c,   \ 
ax  -\-  h'y  =  c .  S 

From  the  first  equation  we  obtain, 

c  —  hy 


(1) 


X  = 


a 


From  the  second  we  obtain, 


X  = 


c  —  Vy 


a 


(2) 


(3) 


Putting  these  two  values  equal,  we  have 

c  —  hy  _  c  —  h'y 


a 


a 


Reducing  and  solving  this  equation  as  in  Chapter  II,  we 

find, 

ac'  —  a'c 

y  = 


'I. ' 


ah'  —  a'h 

which  is  the  required  value  of  y.  Substituting  this  value  of  // 
in  either  of  the  equations  (1),  (2),  or  (3),  and  solving,  we  shall 
find 

~  ahl  —  cih 

If  the  work  is  correct,  the  resi.it  will  be  the  same  in  which- 
ever of  the  equations  we  make  the  substitution.. 


TWO    UNKNOWN  QUANTITIES. 


Ill 


Numerical  Example.    Let  the  equations  be 

x-^l/  =  28,  ) 
3x  —  21/  —  29.  ) 

From  the  first  equation  we  find 

X  =  2S  —  ij, 

_  29  +  2y 


X 


and  from  the  second 

from  whicli  we  have  28  —  ?/  = 


3       ' 

o  J 


w 


2/ =  11. 

Substituthig  this  value  in  the  first  equation  in  x,  it  becomes 

a:  =  28  —  11  =  17. 
If  we  substitute  it  in  the  second,  it  becomes 

29  +  22        51 


X  = 


17, 


3  3 

tlie  same  value,  thus  proving  the  correctness  of  the  work. 


I  '■ 


H 


;    ,,.. 


Eliiniiiation  by  Substitution. 

136.  Rule.  Find  the  value  of  one  of  the  unlaiowiv 
(/naiittties  in  terms  of  the  other  from  either  equation, 
((tid  suhstUute  it  in  the  oth^er  equation.  The  latter  will 
have  but  one  unknown  quantity. 

Example.    Taking  the  same  equations  as  before, 

ax  -{-  hij  ^^  c, 
a'x  +  h'u  =  c', 

tlic  first  equation  gives        x  = 


a 


Substituting  tliis  vahie  instead  of  x  in  the  second  equation, 

it  becomco 

a'c  —  a'bii        , 


a 


Solving  this  equation  witli  respect  to  y,  we  get  the  same 
result  as  before. 


;    ( 


Vr 


.* 


I 


.'   ii 


112 


EQUATIONS   OF   Till':   FIRST   DhVllFF. 


t 


Numerical  Example.  To  solve  in  this  way  the  last  nu- 
merical  example,  we  have  from  the  first  equation  (4), 

x  =  28  —  y. 

Substituting  this  value  in  the  second  equation,  it  becomes 

84  —  3//  —  2?/  ==  29, 

from  which  we  obtain  as  before, 

84  -  20         . , 
y-^    =11. 

This  method  may  be  applied  to  any  pair  of  equations  in 
four  ways  : 

1.  Find  X  from  the  first  equation  and  substitute  its  value 
in  the  second. 

2.  Find  x  from  the  second  ef[uation  and  substitute  its 
value  in  the  first. 

3.  Find  ?/  from  the  ^rst  equation  and  substitute  its  value 
in  the  second. 

4.  Find  y  from  the  second  equation  and  substitute  its 
value  in  the  first. 

Eliiiiiuatioii  by  Addition  or  Subtrtictioii. 

137.  Rile.  Midbiply  each  equation  hij  siirJi  a  factor 
that  the  coefficients  of  one  of  the  unlcnoicn  quantities 
shall  I)ecoj)he  nmnericaUii  equal  in  the  tiuo  equations. 

Tlien,  by  adding  or  subtracting  the  equations,  irr 
shall  have  an  equation  with  but  one  unknown  quantity. 

Rem.  Wc  may  always  take  for  the  factor  of  each  equation 
the  cocfi'cient  of  the  unknown  quantity  to  be  eliminated  in  the 
other  equation. 

Example.    Let  us  take  once  more  the  general  equation 

ax  -{-  by  =  c, 
a'x  -\-  h'y  r=  c'. 

Multiplying  the  first  equation  by  a' ,  it  becomes 

aa'x  +  a'hy  =  a'c. 
Multiplying  tho  second  by  a,  it  becomes 

aa'x  -\-  ah'y  =.  ad. 


TWO    UNKNOWN   QUANTITIES. 


1\M 


The  unknown  quantity  x  has  the  same  cocfllcient  in  tlic 

hist  two  equations.    Subtractinr  them  from  each  other,  we 

obtain 

(a'b  —  ah')  y  ■=  ci'c  —  ac'y 

__  a'c  —  ac' 

Rem.  "We  shall  always  obtain  the  same  result,  whichever 
of  the  above  three  methods  we  use.  But  as  a  general  rule  tlic 
last  method  is  the  most  simple  and  elegant. 

Problem  of  the  Sum  and  Difference. 

The  following  simple  problem  is  of  such  wide  application 
that  it  should  be  well  understood. 

138.  Problem.  Tlte  sum  and  difference  of  two  iium- 
hers  heiii^  given,  to  find  the  nnnibers. 

Let  the  numbers  be  x  and  y. 

Let  s  be  their  sum  and  d  their  difference. 

Then,  by  the  conditions  of  the  problem, 

x-\-  y  =  s, 
X  —  y  :=  d. 
Adding  the  two  equations,  we  have 

2x  =z  s  +  d. 
Subtracting  the  second  from  the  first, 

2y  =  s  —  d. 
Dividing  these  equations  by  2, 


X  = 


s 

+  d 
2 

— 

s 

2 

d 

s 

d 

s 

d 

2 

■■-^ 

2 

2 

y  = 

We  therefore  conclude : 

The  gredtcr  ninuhcr  is  found  by  adding  half  the  dif- 
ference to  half  the  sum. 

The  lesser  niunhcr  is  found  hy  subtracting  half  the 
difference  from  half  the  sum. 
8 


I    I 


-I 


1     I 


■  \ 


I 


1 


114 


EQUATIONS   OF   TUE  FIliST  DEGREE. 


I 
P 


This  result  can  be  illustrated  geoinctriciilly.  Let  AB  and 
BC  be  two  liiK's  placed  end  to  end,  so  that  AC  is  their  sum. 
To  tiiid  their  (llllerence,  we 

cut   oil'   fruni  aU  a  Icnglh         | _f ^ j 

AC  =  BC  ;  then  C'B  is  the 
dillVrence  of  the  two  lines. 

If  P  is  half  way  between  C  and  B,  it  is  the  middle  point 
of  the  whole  line,  so  that 

AP  =  PC  =  |AC  =  ^  sum  of  lines. 
C'P  =  PB  =  tC'B  =  I  dilTerence  of  lines. 

If  to  the  half  sum  AP  avc  add  the  half  ditferenco  PB,  we 
have  AB,  the  greater  line. 

If  from  the  1  alf  sum  AP  we  take  the  half  difference  C'P, 
we  have  left  AC'j  the  lesser  line. 


EXERCISES, 


Solve  the  following  equations: 

1.  3.r  —  9,1/  =  33,     2x  —  3ij  —  18. 

2.  3^:  —  5//  —  13,     2x  +  7y  =  81. 

3.  7^  +  ('>y  =  ^,       Gx  -{■  (')i/  =  b. 

4.  2x  -}-  bi/  =  7)1,      2x  —  ?)!/  =  n. 

5.  ax  -^  hy  —  p,       ax  —  by  =  q. 


8. 


X 


y 


6+7  =^«' 


X 


X 


V 


X 


X 


y 

7 

y 


+  ^  =  18,     5  +  :>   -  29, 


y 


8 
X 


y 
3 


9.     '  {^  +  ^/)  -f  3  {X  -y)  =  102, 
^(•^^  +  ^)-3  0^--v/)  =  G6. 

Note.  Solve  this  oquation  first  as  if  x+y  and  x—y  were  single  sym. 
Lois,  of  which  tlu3  values  are  to  bo  found.  Then  find  x  and  y  by  §  1:>H 
preceding. 


10.     X  +  y  +  {x  —  y)  =  14,    x  +  y  —  [x  —  y) 


10. 


II. 


X 


+  .= 


X 


~-!:rim^  J— T¥"-ijT-:      ri.---.-'7TT^;8n. 


TWO    UNKNOWN  (QUANTITIES. 


115 


Note.    Equations  in  this  form  can  bo  b(\st  solved  as  if  -  and  -  were 
the  uulinovvn  (luuntities.     See  next  exercise.  ^ 


3       2 

12. 

X      y 


n      4       5 
10'    x'^  y 


n 
O. 


Soi-UTION.    If  we  multiply  tl»e  first  equation  by  4,  and  the  second  by 
;j,  we  have 

12  _   8    _  44  _  22 


X 

12 

15 

+ 

X 

!/ 

10 


9  = 


45 


Subtracting  the  first  from  the  second,  we  havo 

23  _  23 

2/   -   5  ' 

y  =  5. 
Again,  to  eliminate  - ,  we  multiply  the  first  equation  by  5  and  the 

if 


whence, 


second  by  2  and  add.    Thus, 

15       10 


X 


y 

10 


n 

2  ' 


8^10        _        12 


23 

X 


23 


2  ' 


whence, 


X  —  2. 


n- 

2      3        7       2      3  _        1 
x'^  y~  \2'    X      y            12 

14. 

12        5       2       1         5 

x"^  y       12'    X      y       24 

15- 

5       3             13       11 

X      y  ~       6'    X      y  ~  'SO 

16. 

5              3                 13 

X  -\-l      y      1"       e'.-r  +  l 

y 

1 
—  1 

1 

30 

17- 

2              3            7          2 
X  +  2   '    v-3  ~  12'     x  +  2 

3 

y- 

3  ~ 

1 
12 

!     i 


-  h 


110 


EqUATIONS  OF  THE  Flll.ST  DEGHEK 


1 8.         +      =  c,         —      =  a. 

'      X  —  ij  a  4-  3 

a  -\-  h      a  —  h  4ab 


=  1. 


-1 

'1 


Case  II.  Equations  of  the  Fh'st  Drffvee  irith 
Three  or  3Iore  Unknown  Qttantfties, 

l.*51).  Wlicn  tlie  values  of  sevcriil  iiiiknowti  (jiiMiititics  aic 
to  be  found,  it  is  necessary  to  luive  as  many  equatious  as  un- 
known ([uantities. 

If  tliere  are  more  unknown  quantities  tiian  equations,  it, 
will  be  impossible  to  determine  the  values  of  all  of  them  from 
the  equations.  All  that  can  be  done  is  to  determine  the  vahie 
of  some  in  terms  of  the  others. 

If  the  number  of  equations  exceeds  that  of  unknown  quan- 
tities, the  excess  of  equations  will  be  superfluous.  If  there 
arc  71  unknown  quantities,  their  values  cau  be  found  from  any 
n  of  the  equations.  If  any  selection  of  n  eciuations  we  choose 
to  make  gives  the  same  values  of  the  unknown  quantities,  the 
equations,  though  sUj»erfluous,  will  be  consistent.  If  ditfereuL 
values  are  obtained,  it  wmII  be  impossible  to  satisfy  them  all. 


Elimination. 

140.  When  the  number  of  unknown  quantities  exceeds 
two,  the  most  convenient  method  of  elimination  is  generally 
that  by  addition  or  subtraction.  The  unknown  quantities  arc 
to  be  eliminated  one  at  a  time  by  the  following  method  : 

I.  Select  an  unlcnoivn  quantity  to  he  first  eliminated. 
It  is  best  to  he^in  with  the  quantity  which  appears  in, 
the  fewest  equations  or  has  the  simplest  coefficients. 

II.  Select  one  of  the  equations  containing  this  un- 
known quantity  as  an  eliminating  equation. 

III.  Mifninate  the  quantity  between  this  equation 
and  each  of  the  others  in  succession. 


TPRKE   OR    MOR/'J    UXhWOWA    I^UANTITIHS.       117 


We  shiill  then  liavo  a  secoiul  system  of  eqiuitioiis  le?s  by 
our  ill  iiumlter  than  llio  orii^iiuil  HVstL'in  uiul  contain iii;^'  a  num- 
hcr  of  iiiikiiuwii  (juaiitities  one  k'ss. 

I V.  J\('}K'(ft  the  process  on  the  nrtr  sj/sfmi  of  rt/u/ffions, 
(tnd  roitfiufie  the  vcprtition  until  on/ijone  cf/Kf/fion  with 
our  iinkiioirii  qiKintitij  is  left. 

V.  Jlorin^  found  the  vulue  of  this  hist  uuIkuowu. 
(/uftntitf/,  the  vulues  of  the  others  run  tte  founil  hij  suc- 
eessire  sultsiitution  in  one  equation^  of  rueh  system. 

Example.     Solve  the  equations 

(1)  4:X  —  '.\ii—    z  ^    n—  r  =  0, 

(:))  X  —   y  -\-  'Zz  '\-  'III  —  \0  —  0, 

(;5)  'Hx  +  'ly  —    z~2n  —  2  =  0, 

(4)  X  -\-  2y  -^    z  -}-    u  —  1!)  =  0. 


('0 


Wo  shall  Bolpct  .?' as  tlif  first  quantity  \n  be  climiiiatfMl,  and  tnkt»  the 
last  fciuation  as  the  (.'liniinatinf^  one.  We  first  multiply  this  (Mjuation  by 
threo  such  factors  that  the  coefficient  of  x  shall  become  ecjual  to  the  co- 
etliciont  of  x  in  each  of  the  other  equations.  These  factors  are  4,  1,  and  2. 
We  write  the  products  under  each  of  the  other  (-(luations,  thus  : 


Eq.  (1), 

(4)  X  4, 

E(i.  (2), 
(4)  X  1, 

Kq.  (3), 
(4)  X  2, 


4.r  —  3//  —    .ir  +    ii  —    7  =  0, 

4:X  +  8//  +  4z  +  4u  —  76  =  0. 

X—    y  +  2z  -f  2ii  —  10  =  0, 

X  -h  2y  -\-    z  -{-    u  —  \U  =  0. 


z  —  2i( 


2x-\-2y- 

2x  +  4//  -^  2z  -\-  2u 


2=0, 
38  =  0. 


By  subtracting  the  one  of  each  pair  from  the  other,  we  obtain  the 
equations, 

11//  +  5^  +  Su  —  G9  =  0,  ) 

—    z—    n  —    9  =  0,  V 

3G  =  0.  ) 


3// 

2y  -^^z  -\-  4:U 


(*) 


The  unknown  quantity  x  is  here  eliminated,  and  we  have  three  equa- 
tions with  only  three  unknown  quantities.  Now  eliminating  ?/  by  means 
of  the  last  equation,  in  the  same  way,  and  clearing  of  fractions,  we  find 
the  two  equations, 

23z  +  38?^ 

Uz  -\-  Uu 


-  258  =  0,  ) 

—  90  =  0.  j 


(c) 


ri\ 


118 


El^VATlONH   OF   THE  FlliST  DEiilWK, 


'J'lic  pniltlcni  is  now  reduced  to  two  eciimtions  with  two  unknown 
qiumtitie?*,  which  we  huve  aln-iuly  nhowu  how  to  boIvu.  Wo  lind  hy 
holvin^  them, 

«  =   —  ^, 

u  —  8. 

Wo  next  find  the  value  of  y  by  Huhstirutln^'  these  vnlueH  of  «  and  u 
in  eitlier  of  tlie  e<|iuitionH  yh).     The  tir.st  of  tlieni  tlius  heconien: 


from  wliich  we  find, 


11//—  10  +  '.U  — 01)  =  0, 


^^'e  now  Hul)stitntc  the  values  of  y.  z,  and  u  in  either  of  equations  {a). 
Die  ^ie(•oIl(l  of  tlu>  latter  liecoines 

X  _  T)  —  4  +  10  —  10  —  0, 

and  tlie  fourth  becotne.s, 

X  +  10  —  'I  +  8  —  10  =  0, 

cither  of  which  ^^ivoH 

X  =  3. 

We  ran  now  prove  the  results  by  substituting:  tlie  values  of  T,i/,z, 
and  u  in  all  four  of  (M|uationH  iif),  and  seeinj:^  whether  they  are  all  satibfied, 

EXERCISES. 

1,  One  of  the  ])cst  oxorciscs  for  tlie  student  will  lie  thnt  of 
resolvini,^  tiie  previous  eqiuitioiis  {(()  by  tiikiM<jj  the  jjist  e((ua- 
tion  as  tlie  eliininatiii.u^  one,  iiiid  perforniini^  tiie  eliniiuatioii 
in  (liU'erent  orders:  that  is,  beo;in  by  eliniiiiatinr^  tf,  then 
repeat  the  Avholc  process  beginning  with  z,  etc.  The  linal 
results  will  always  l)C  the  same. 

2.  Find  the  values  of  x^,  Xc,,  x^,  and  x^,  from  the  equa- 
tions, 


X, 


•Tg    -f~  '^3  '^4    —       0, 


X, 


This  exam]ile  requires  no  multiplication,  but  only  addition  and  sub. 
traction  of  the  dill'orent  c(iuation.s. 

3.  2x -{-  by  -i- 3z  =  13, 

2x  -\-  2y  —   z  =^  I2j 

bx  +  by  —  2z  =  20. 


rwniLKMs. 


liu 


3«  +  'in 

•'i// 

=   IS, 

•,)x  +  ;/ 

— 

4?* 

=    0, 

x^lz 

♦'// 

=  .'J.'), 

JSz^^x  —  8y 

-f- 

'Zu 

=  15. 

X  +  y  -\-  z  =z  a, 
y  -\-  z  -{-  n  =  b, 

6. 

1        1 

—      =  m 
X      y 

1       1 

Z  -^  U  ■{■  X  =.  c, 
u  -\-  x  -\-  y  =:  d. 

1       1 

-  +      =  p. 

z       X       '■ 

<,i 


PROBLEMS    FOR    SOLUTION. 

1.  A  mjin  liiul  a  siuldle  wortii  $1')  and  two  lioi^cs.  If  tlu? 
siiddlo  1)0  put  oil  liorse  A  lie  will  bo  douhk'  the  value  of  B,  hut 
if  it  he  put  on  B  ins  vahu;  will  he  etiual  to  that  of  A.  What 
is  the  value  of  each  horse  ? 

2.  What  number  of  two  digits  is  equal  to  7  times  the  sum 
of  its  digits,  and  to  *J  times  the  ditference  of  its  digits  increased 
hy  4? 

Let  X  be  tho  first  digit,  or  the  numl)cr  of  tons,  and  y  the  unitH.  Then 
tlio  nmulH^r  itsolf  will  ))o  10j;+y.  Seven  times  the  sum  of  tho  digitH  uro 
Ix  +  li/,  and  I)  times  the  difference  is  \){x—p  +  4). 

3.  A  number  of  two  di.cfits  is  equal  to  f]  times  the  sum  of 
its  digits,  and  if  9  be  subtracted  from  the  number  the  digits 
are  reversed.     What  is  the  numljer? 

4.  Find  a  number  of  two  digits  such  that  it  shall  be  eqmil 
to  0  times  the  sum  of  its  digits  increased  bv  1,  while  if  18  bo 
subtracted  from  the  number  the  digits  will  Le  reversed. 

5.  Find  a  number  which  is  greater  Ijy  2  than  5  times  the 
sum  of  its  digits,  and  if  9  bo  added  to  it  the  digits  will  be 
reversed. 

6.  What  number  is  that  which  is  equal  to  9  times  the  sum 
of  its  digits  and  is  4  greater  than  11  times  their  dilTerence? 

7.  What  fraction  is  that  which  l)ecomes  e([Uid  to  |  when 
^he  numerator  is  increased  by  2,  and  ecjual  to  ^-  when  the  de- 
nominator is  increased  by  4. 

8.  Two  drovers  A  and  B  went  to  market  with  cattle.  A 
sold  50  and  then  had  left  half  as  many  as  B,  who  had  sold 
none.  Then  B  sold  54  and  had  remaining  half  as  many  as  A. 
How  many  did  each  have  ? 


M 


120 


EQUATIONS  OF  Till-:  FIRST  DEGREE. 


g.  A  l)oy  bought  42  apples  for  a  dollar,  giving  3  cents  each 
for  the  good  ones  and  2  cents  each  for  the  poor  ones.  How 
nuiny  of  each  kind  did  ho  l)U,y? 

10.  Find  a  fraction  which  becomes  equal  to  ^  when  its 
denoniinuror  is  increased  by  1.'3,  aiul  to  f  when  4  is  subtracted 
from  its  numerator. 

11.  Find  a  fi'aciion  which  will  become  equal  to  |  l)y  addinu' 
2  to  its  uumerator,  "r  by  adding  to  its  denominator  o,  will  hi  - 
come  ^. 

12.  A  huckster  bought  a  certain  number  of  chickens  at 
32  cents  each  and  of  turkeys  at  75  cents  each,  paying  814  for 
the  whole,  lie  sold  the  chickens  at  48  cents  each,  and  the 
turkeys  at  11  each,  realiziug  820  for  the  whole.  How  many 
chickens  and  how  many  turkeys  had  he  ? 

13.  An  applewoman  bought  a  lot  of  a]>ples  at  1  cent  each, 
and  a  lot  of  pears  at  2  cents  each,  paying  81. lO  for  the  whole. 

11  of  the  apples  and  7  of  the  pears  were  bad,  but  she  sold  the 
good  a})})les  at  2  cents  each  and  the  good  pears  at  3  cents  each, 
realizing  82.00.     How  many  of  each  fruit  did  she  buy? 

14.  When  Mr.  Smith  was  married  lie  Avas  |  older  than  his 
u'ife  ;  twelve  years  afterward  he  was  \  older.  What  were  their 
ages  when  married  ? 

15.  A  and  B  together  can  do  a  piece  of  Avork  in  0  dnys,  but 
A  working  alone  can  do  it  0  davs  sooner  than  B  working; 
alone.     In  what  time  could  each  of  them  do  it  singly  ? 

16.  A  husband  being  asked  the  age  of  himself  and  wife, 
replied:  "If  you  divide  my  age  G  years  hence  by  her  age 
0  years  ago,  the  Cfuotient  will  be  2.     But  if  you  divide  her  age 

12  3'ears  hence  l)y  mine  21  years  ago,  the  ((uotient  will  be  5. 

17.  The  sum  of  two  ages  is  1)  times  their  difference,  Inif 
seven  years  ago  it  was  only  seven  times  their  difference.  What 
are  the  ages  now  ? 

18.  Two  trains  set  out  at  the  same  moment,  the  one  to  go 
i'rom  Boston  to  Springfield,  the  other  from  Springfield  to  )3os- 
ton.  The  distance  behveen  the  two  cities  is  OS  miles.  They 
meet  each  other  at  the  end  of  1  hr.  24  min.,  and  the  train  fri»m 
Boston  travels  as  far  in  4  hrs.  as  the  other  in  3.  What  was  the 
speed  of  each  train  ? 

19.  A  ^'■rocer  bought  50  lbs.  of  tea  and  100  lbs.  of  coffee  for 
oO.     He  sold  the  tea  at  an  advance  of  ^  on  his  price,  and  the 

coffee  at  an  advance  of  j^,  realizing  877  from  both.  At  what 
price  per  pound  did  he  buy  and  sell  each  article  ? 

Note,  If  x  and  y  are  the  prices  at  which  he  bought,  then  Ja*  and  |y 
are  the  prices  at  which  he  sold. 


INCONSISTENT  EQ  UA  TIONS. 


121 


20.  For  p  dollars  I  can  piuTbase  eitlier  a  pounds  of  tea  and 
h  pounds  of  cott'ee,  or  ))i  jjounds  of  tea  and  ii  i)ounds  of  colfeo. 
What  is  tlie  i)rice  })er  pound  of  each  ? 

21.  A  goldsmith  had  two  ingots.  The  first  is  composed  of 
equal  parts  of  gold  and  silver,  while  the  second  contains  5  parts 
of  gold  to  1  of  silver.  He  wants  to  take  from  them  a  watch- 
case  having  4  ounces  of  gold  and  1  ounce  of  silver.  How 
much  must  he  take  from  each  ingot  ? 

2  2.  A  banker  has  two  kinds  of  coin,  such  that  a  pieces  of 
the  first  kind  or  b  i)ieces  of  the  second  will  make  a  dollar.  If 
he  wants  to  select  c  i)ieces  which  shall  be  worth  a  dollar,  how 
many  of  each  kind  must  he  take  ? 

23.  A  has  a  sum  of  money  invested  at  a  certain  rate  of 
interest.  B  has  81000  more  invested,  at  a  rate  1  per  cent, 
jiigher,  and  thus  gains  $80  more  interest  than  A.  C  has  in- 
vested -i^oOO  more  than  B,  at  a  rale  still  hif^her  l)y  1  per  cent., 
and  thus  gains  $70  more  than  B.  What  is  the  amount  each 
person  has  invested  and  the  rate  of  interest  ? 

24.  A  grocer  had  three  casks  of  wine,  containing  in  all 
311  gallons.  He  sells  50  gallons  from  the  first  cask ;  then 
pours  into  the  first  one-third  of  what  is  in  the  second,  and 
then  into  the  second  one-fifth  of  what  is  in  the  third,  after 
which  the  first  contains  10  gallons  more  than  the  second, 
and  the  second  10  more  than  the  third.  How  much  wine  did 
each  cask  contain  at  first  ? 


r 


I      I 


>' 


Equivalent  and  Inconsistent  Equations. 

141,  It  is  not  always  the  case  that  values  of  two  unknown 

quantities  can  be  found  from  two  equations.     If,  for  example, 

we  have  the  equations 

X  -\-'2ij  =z  3, 

2.r  +  1?/  :=  G, 

we  see  that  the  second  can  be  derived  from  the  first  by  multi- 
plying both  members  by  2.  Hence  every  pair  of  values  of  x 
and  y  which  satisfy  the  one  will  satisfy  the  other  also,  so  that 
the  two  are  equivalent  to  a  single  one. 

If  the  equations  were 

a;  4-  2?/  =  5, 

2a:  +  4y  =  6, 

there  would  be  no  values  of  x  and  y  which  would  satisfy  both 
equations. 


't 


123 


EQUATIONS   OF   TUE  FIRST  DEGREE. 


%. 


(1) 


For,  if  \VQ  multiply  the  first  by  2  and  subtract  the  second 
from  the  product,  we  shall  liave, 

Ist  C((.   X  'I,                     2x  +  4?/  =  10 
2d  eq. ,  2x  +  4//  z= (> 

Remainder,  0  ==    4, 

an  impossible  result,  which  shows  that  the  equations  are  incon- 
sistent. This  will  be  evident  from  the  equations  themselves, 
because  every  pair  of  values  of  x  and  ;/  vvnich  gives 

2x  +  4?/  =  G, 

must  also  give  a*  +  2^/  —  3, 

and  therefore  cannot  give   x  -\-  2i/  =^  5. 

142.  Generalization  of  the  preceding  result.  If  we  take 
i.ny  two  equations  of  the  first  degree  between  x  and  ij  which 
we  may  represent  in  the  form 

ax  +  hy  =  c,   ) 
a'x  +  b'lj  =  c',  \ 

and  eliminate  x  by  addition  or  subtraction,  as  in  §  137,  we  have 
for  the  equation  in  y, 

(a'b  —  ah)  y  =  a'c  —  ac'. 

Now  it  may  happen  that  we  have, 

a'b  —  ab'  =  0  identically.  (2) 

In  this  case  y  will  disappear  as  well  as  x,  and  the  result 

will  be 

a'c  —  ac'  ~  0. 

If  this  equation  is  identically  true,  the  two  equations  (1) 
will  be  equivalent ;  if  not  true,  they  will  be  inconsistent.  In 
neither  case  can  we  derive  any  value  of?/  or  x. 

If  we  divide  the  above  equation,  (2),  by  aa!  we  shall  have 

h       V 

a       a' 
Hence, 

TJicorem..  If  the  quotient  of  the  coefficients  of  the 
unknown  quantities  is  tlie  same  in  the  two  equations, 
they  will  be  either  equivalent  or  inconsistent. 


INEQUALITIES. 


123 


This  tlicorem  can  l)c  expressed  in  (lie  foUowiiif^  form: 

If  the  terms  contai/u/ig  the  uiilawii'ii  r/itfuititi/  in  the 
one  eqitatioii  cnn  he  j}iitlti/)Iie(/  hi/  suc/i  n  fiietor  that 
{hcij  sJndl  hotli  heeonie  eqiidl  to  tlic  eorres/jondino'  /r;7;?.s' 
of  tJie  otlier  eijuatioib,  the  tico  equations  will  he  either 
('(jitivalent  or  iiicoiisistent. 

Proof.  If  there  be  sucli  a  factor  m  that  multiplying  the 
first  equation  (1)  by  it,  we  shall  have 

ma  ±=i  a, 
nib  =z  b'. 
Eliminating  m,  we  find 

a'h  —  ab'  =  0, 
the  criterion  of  inconsistency  or  equiv.alence. 

143.  When  two  equations  are  inconsistent,  tliere  are  no 
values  of  the  unknown  quantities  which  will  satisfy  both  equa- 
tions. 

When  they  are  e(|uivalent,  it  is  the  same  as  if  we  had  a 
single  equation ;  that  is,  we  may  assign  any  valne  avc  })lease  to 
one  of  the  unknown  quantities,  and  iind  a  corresponding  value 
of  the  other. 


-♦-<-♦- 


CHAPTER     IV. 

OF     INEQUALITIES. 

144.  Def.  An  Inequality  is  a  statement,  ia  tlie 
language  of  Algebra,  that  one  quantity  is  algebraically 
greater  or  less  than  another. 

Def.  The  quantities  declared  unequal  ai'<3  called 
Members  of  the  inequality. 

The  statement  that  A  is  greater  than  B,  or  that  A  —  B  is 

positive,  is  expressed  by 

A  >  B. 


124 


INEQUAJJTIES. 


Tlijit  A  is  less  than  B,  or  that  A  —  B  19^  negative  is 
expressed  by 

A  <^  B. 

The  form  ^  >  ^  >  C 

indicates  that  the  quantity^  is  less  tlian  A  but  greater  than  ( . 

The  form  A^  B 

indicates  tliat  A  may  be  either  equal  to  or  greater  than  B,  bui 
cannot  be  less  than  B. 

Properties  of  Inequalities. 

145.  Theorem  I.  An  inequality  will  still  subsist 
after  the  same  quantity  lias  been  added  to  or  subtracted 
from  each  member. 

Proof.  If  the  inequality  be  A  ^  B,  A  —  B  must  be  posi- 
tive. If  we  add  the  same  quantity  //  to  A  and  B,  or  snbtnu  1 
it  from  them,  we  shall  have  A  ±^  If — (l^^II),  which  i- 
equal  to  A  —  B,  and  therefore  positive.     Hence,  if 

A>  B, 

then  A±Hy  B  ±11. 

Cor.  Tf  any  term  of  an  inequality  be  transposed 
and  its  sign  changed,  the  inequality  will  remain  true. 

TJieorem  II.  An  inequality  will  still  subsist  after 
its  members  have  been  multiplied  or  divided  by  the 
same  positive  number. 

Proof,  li  A  —  B  is  positive,  then  {m  or  n  being  positive) 
m  (A  —  B)  or  mA  —  mB  will  be  positive,  and  so  will 

B 

n 


A-B           A 

or    — 

01                  n 

Hence,  if 

A>  B, 

then 

w?J.  >  mB 

and 

n       n 

It  mn 

mA  —  m 

TJieo 
nuiltipli 
the  dire( 

That 

then 


iind 


Theo\ 
several 
member 

Tlieo\ 
subtract 
another, 
of  the  la 

That 

then 

The  pn 
plied  by  th 

Theo', 
equalitj'- 
still  subi 

Proof 

Becau 
(Th.  II), 

Also, 

by  ^, 


INEQUALITIES. 


125 


It  may  be  shown  in  tlie  same  way  that  if  m  or  n  is  negative, 

A       B 

viA  —  )nB  or will  be  negative.     Hence, 

n       n  ° 

T/ieorem  III.  If  both  members  of  an  inequality  be 
multiplied  or  divided  by  the  same  negative  number, 
the  direction  of  the  inequality  will  be  reversed. 

That  is,  if  A  >  B, 

then  —  mA  <  —  mB, 

n  ^        n 


aiHi 


Theorem  IV.  If  the  corresponding  members  of 
several  inequalities  be  added,  tlie  sum  of  the  greater 
members  will  exceed  the  sum  of  the  lesser  membei's. 

Tlieorein  Y.  If  the  members  of  one  inequality  be 
subtracted  from  the  non-corresponding  members  of 
another,  the  inequality  will  still  subsist  in  the  direction 
of  the  latter. 


That  is,  if 


then 


Ay  B, 

^>  y, 

-y  y  B  —  x. 


The  proof  of  the  last  three  theorems  is  so  simple  that  it  may  be  sup- 
plied by  the  student. 

Theorem  VI.  If  two  positive  members  of  an  in- 
equalit}^  be  raised  to  any  power,  the  inequality  will 
still  subsist  in  the  same  direction. 

Proof.     Let  the  ineqnality  be 

A>  B.  (a) 

Because  A  is  positive,  we  shall  have,  by  multiplying  by  A 

(Th.  II), 

A^>AB.  (1) 

Also,  because  B  is  positive,  we  have,  by  multiplying  (a) 

AB  >  i?2.  (2) 


-  h' 


\  '  I 


.■!   '     '.I 


•:]• 


i  if! 


120  INEqUALITIES. 

Therefore,  from  (1)  and  (^), 

A''  >  7i2,  (3) 

Mu]t;-^ivmg  the  last  inequality  by  A, 

A^  >  Air^.  (.1) 

Miiltii)lying  (2)  by  i^, 

yiJ52  >  BK  {:>) 

Whence,  A^  >  M 

Tlie  process  mr^y  be  continued  to  any  extent. 

Examples  of  the  Use  of  Inequalities. 

140.    Ex.  I.  If  «  and  h  be  two  positive  quantities,  such 
that 

a"^  +  1)^  -  1, 

we  must  have  a  -\-  h  y  1. 

Proof.    If  «  +  /^  <  1, 

we  should  liave,  by  squaring  the  members  (Th.  VI), 

a^  -^ 'lab  ^  U^  ^\ ', 
and  by  transposing  the  product  ^ab  (Th.  I,  Cor.), 

6'^  +  Z»2  ^  1  —  "lab. 

Because  a  and  b  are  positive,  'lab  is  positive,  and 

1  —  lab  <  1. 

Therefore  we  should  have 

«2  -f  ^*2  <-  1^ 

and  could  not  have  c^  -^-W-  =z\,  as  was  originally  supposed. 
Ex.  2.    If  a,  b,  m,  and  n  are  positive  quantities,  such  that 


am 
b-^n 


(^0 


a  A~  771 

then  the  value  of  the  fraction will  be  contained  between 

a  +  71 


771 


the  values  of  y  and  —  ;   that  is, 
0  n 


Top 


is  positi^ 


Fron 
by  the  p 

That 
with  thii 
as  assert 

The 


I.  P 

j;ero,  anc 


2. 

P] 

3- 

If 

4- 

If 

5- 

If 

6. 

If 

7. 

If 

8.  If 
then  ab 

SUGG^ 

negative. 

INEqUALITIES. 


127 


a       a  ■{-  m       m 

I       b  +  n         n 


(1) 


C'i) 


(:0 


To  prove  tlie  first  iiie([uulity,  we  must  .show  that 

a      It  +  in 
b      b  -{•  n 

is  positive.     Redueing  this  expression  by  §  lOG,  it  becomes 

an  —  bin 

Vi/TTn)' 

From  the  original  inequality  (a)  we  have,  by  multiplying 
by  the  positive  factor  bn, 

an  y  bm. 

That  is,  an  —  bm  is  positive  ;  therefore  the  fraction  (:}) 
with  this  positive  numerator  is  also  positive,  and  (2)  is  iJositi\  e 
as  asserted. 

The  second  inequality  (1)  may  be  proved  in  the  same  way. 

EXERCISES. 

1.  Prove  that  if  a  and  b  be  any  quantities  different  from 
j:ero,  and  1  >  a;  >  —  1,  we  must  have 

a2  _  2abx  +  ^(2  >  0. 

2.  Prove  that  ( -^^ — I  y  ab. 

3.  If  ^x  —  5  >  13,  then  x  >  G. 

4.  If  (jx  >  ^^  +  18,  then  x  >  4. 

5.  If  ~  -  5^  >  ^  -  3,  then  xyb. 

J) ffi 

6.  If  m  —  nxyp  —  qx,   then   x  >  ~ 

^       ^  q  —  n 

7.  If  ^ <  1 J  and  wi  and  y  of  like  sign  :  x  <  y. 

'my  ./  c:>  . 

8.  If  rt^  +  J2  _|_  c2  r=  1,  and  a,  b,  and  c  are  not  all  equal, 
then  ab  -{■  be  +  ca  <  1. 

Suggestion.  The  squares  of  a  —  h,h  —  c,  and  c  —  a  cannot  Lo 
negative. 


BOOK    IV. 
A^ A  TI O     A  ND     P ROPO R  TIO N, 


'^^ 


CHAPTER    I. 
NATURE    OF    A     RATIO. 

147.  Def.  The  Ratio  of  a  quantity  A  to  another 
quantity  ^  is  a  number  expressing  thi;  value  of  A  when 
compared  with  B  as  the  standard  or  unit  of  measure. 

Examples.     Comparing        a 

tiie  lengths  A,  B,  C,  D,  it  ^^ 

will  be  seen  that  '     ' 

A  is  '^\  times  D\  ^ 

B  is  \  of  Z>;  2)    |     |     |     ,     | 

C  is  I  of  D. 

We  express  this  relation  by  saying, 

9 
The  ratio  of  A  io  D  \%  %\  or  -.\ 


I     i 


I     I     I 


<< 


it 


''     B  to  D  is 
''      C  to  D  is 


1 

2' 

3 

—  • 

4 


(1) 


148.  The  ratio  of  one  quantity  to  another  is  expressed  l)y 
writing  the  unit  of  measure  after  the  quantity  measured,  and 
inserting  a  colon  between  them. 

The  statements  (1)  will  then  be  expressed  thus  : 

A:D  =  ^  =  p      ^''^  =  \'^       ^''-^  =  1' 

Def.  The  two  quantities  compared  to  form  a  ratio 
are  called  its  Terms. 


RATIO. 


129 


Def.  The  quantity  measured,  or  the  lirst  term  of 
the  I'litio,  is  calh'd  tlie  Antecedent. 

The  unit  of  measure,  or  tlie  si'cond  term  of  tlie  ratio, 
is  called  the  Consequent. 

liKM,  When  the  antecedent  is  greater  tluiii  tiie  conse4iient, 
I  lie  ratio  is  greater  than  unity. 

When  tlie  antecedent  is  less  than  the  consequent,  the  ratio 
is  less  than  unity. 

149.  To  find  the  ratio  of  a  qnantity  A  to  a  standard  Uy 
we  imagine  ourselves  as  measuring  otf  the  (juantily  .1  with  (7 as 
a  carpenter  measures  a  board  with  his  foot-rule. 

There  arc  then  three  cases  to  be  considered,  according  to 
the  way  th    measures  come  out. 

Case  1.  AVe  may  find  that,  at  the  end,  A  conu  out  an 
exact  numl)cr  of  times  U.  The  ratio  is  then  a  whole  number, 
and  we  say  that  U  exactly  measures  A,  or  that  A  is  a 
multiple  of  U. 

Case  II.  We  may  find  that,  at  the  end,  the  measure  does 
not  come  out  exact,  but  a  i)iece  of  A  less  than  U  is  left  over. 
Or,  .1  may  itself  be  less  than  U.  We  must  then  llnd  what 
traction  of  U  the  piece  left  over  is  equal  to.  This  is  done  by 
dividing  U  up  into  such  a  number  of  equal  parts  that  one  of 
these  parts  shall  exactly  measure  A  or  the  piece  of  A  wiiich  is 
left  over.  The  ratio  will  then  be  a  fraction  of  which  the  num- 
ber of  parts  into  which  U  is  divided  will  be  the  denominator, 
and  the  number  of  these  parts  in  A  the  numerator. 

Example.  If  we  find  that 
by  dividing  U  into  7  parts,  -i  of 
these  parts  will  exactly  make  A, 
I  lien  A  =.^  oi  U,  and  we  have  for  the  ratio  of  A  to  CT, 

A  '.U-^  Y 

If  we  find  that  A  contains  U  3  times,  and  that  there  is 
then  a  piece  equal  to  f  of  [7  left  over,  we  have 

25 


I     I     I     I     I 


=  U 
=  A 


u  =  ^=  ^ 


-t  !  ! 


I  |.  i  ■  I , 


|i  fl 


9 


.'     ti! 


130 


HAT  JO. 


f 


^^ 


The  3  U*ii  aro  equal  to  ^^^  of  U,  so  that  wc  may  also  say 


A 


,  of  U,     or     A  :  U  = 


wliich  is  simply  the  result  of  reducing  the  ratio  ']^  to  an  iin- 
j  in  (per  traction. 

Ill  general,  if  wo  llnd  that  by  dividing  U  into  n  jiarts,  J 
will  be  exactly  ///  of  these  parts,  then 


A 


n 


whether  in  is  greater  or  Ics'^  than  n. 

When  the  nnignitude  of  A  measured  by  U  can  be  exactly 
expressed  by  a  vulgar  IVaction,  A  and  U  are  said  to  be  com- 
mensurable. 

OAvSE  III.  It  may  happen  that  there  is  no  number  or  frac- 
tion which  will  exactly  express  the  ratio  of  the  two  magnitudes. 
The  latter  are  then  said  to  be  incommensurable. 

150.  Theorem.  The  nitio  of  two  incommensurable 
magnitudes  may  always  l)e  expressed  as  near  the  true 
value  as  we  please  by  means  of  a  fraction,  if  we  only 
make  the  denominator  large  enough. 

Examples.  Let  us  divide  the  unit  of  measure  into  20 
parts,  and  suppose  that  the  antecedent  contains  more  than  :28 
))ut  less  than  20  of  these  parts.  Then,  by  supi)osing  it  to  con- 
tain 28  parts,  the  limit  of  error  will  be  one  part,  or  -^  of  the 
standard  unit. 

In  general,  if  we  wish  to  express  the  ratio  within  1  n^^  of 
the  unit,  we  can  certainly  do  it  by  dividing  the  unit  into  n  or 
more  parts,  or  by  taking  as  the  denominator  of  the  fraction  a 
number  not  less  tiuni  n. 

Illustration  l)y  Decimal  Fractions.  The  square  root  of  2 
cannot  be  rigorously  expressed  as  a  vulgar  or  decimal  fraction. 
But,  if  we  suppose 

•\/2  =  1.4      =  \^,    the  error  will  be  <  jV  5 

a/2  —  1  414  —  li-iA      ''  "      <r'  — 1— 

etc.        etc.  etc.  etc. 


we  mear 


NATURE  OF  A    RATIO. 


131 


Since  the  docimalH  may  bo  continuod  withoul  end,  ilio 
>(|iuirc'  root  of  'I  can  l)t!  cxpiviised  as  a  dccinnd  fraction  with  lui 
crrov  Ic'HS  than  any  us.signublc  (juaiitily.  This  general  laet  id 
expressed  hy  -saying: 

Tkc  limit  of  f/ie  error  irltick  ice  make  hij  representing 
aiiy  iiicoimnensardblo  ratio  as  a  fraetioii  is  zero. 

151.  lidfio  as  a  QhoHchI.  From  Case  II  and  the  cxi)lana- 
tions  which  precede  it  we  see  that  when  we  say 

4 


A  :  U  = 


V 


we  mean  the  same  thing  as  if  we  had  said, 

yl  is  4  of  U,    or    .1  =  4  C7. 

If  A  and  U  are  mimbcrs,  we  may  divide  both  sides  of  this 
L'ljuation  by  U,  and  obtain, 

A  -i 

u  ~  1 

We  therefore  conclude  that  when  A  and  U  are  numlicrs, 
That  is,  A  '.U  =  j^' 

Theorem.  The  ratio  of  two  numbers  is  equal  to  the 
quotient  obtained  by  dividing  the  antecedent  term  by 
tlie  consequent. 

In  the  case  of  magnitudes,  tlie  relation  of  a  ratio  to  a  quo- 
tient may  be  shown  thus  : 

Let  us  have  two  magnitudes  M  and  V,  such  that  M  is 
4  times  V.     Then  we  may  write  the  relation, 

M  =  4F. 

Dividing  by  4,  we  have 

4   -  ^• 

Since  F  is  not  a  number,  wt  cannot,  strictly  speaking, 
multiply  or  divide  by  it.  But  we  may  take  the  ratio  of  M  to 
F  without  regard  to  number,  and  thus  find, 

M  '.  F  =  4. 


132 


RATIO. 


■  ;  >l( 


Rem.  The  theory  of  ratios  the  termtj  of  which  are  magni- 
tudes and  not  nund)ers,  is  treated  in  (Geometry. 

In  AljLjehra  we  consider  the  ratios  of  numbers,  or  of  nuig- 
iiitudes  represented  )»y  numbers. 

15*^.  Def.  If  w(i  intorcliange  the  terms  of  a  ratio, 
the  result  is  called  tlie  Inverse  ratio. 


If 

then 


That  is,   U '.  A  ia  the  inverse  oi  A  '.U, 

m 


U'.  A 


n 


U  =   —  A, 

n 
m 


and  wc  have,  by  dividing  l)y      , 

7h 


A  =  '^U, 
m 


or 


A  :  V 


n 
m 


71  Tfl 

Because  —  is  the  reciprocal  of  — ,  wc  conclude : 

m  ^  n 

Theorem.  The  inverse  ratio  is  the  reciprocal  of  the 
direct  ratio. 

Properties  of  Ratios. 

153.  Tlieorem  I.  If  both  terms  of  a  ratio  be  multi- 
plied by  the  same  factor  or  divided  by  the  same  divisor, 
the  ratio  is  not  altered. 

Proof.     Katio  of  ^  to  ^  =  ^  :  ^  = 
If  m  be  the  factor,  then 

Ratio  of  mB  to  w^  =  mB  :  mA  ■■ 
the  same  as  the  ratio  of  B  to  A, 


B 
A 


mB  _B 

mA  ~  A ' 


154.  Tlieorem  II.     If  both  terms  of  a  ratio  be  in- 
creased by  the  same  quantity,  the  ratio  will  be  increased 


if  it  is 
(hat  is 

EXA 

1  til  botli 
m  rit'W  of 

lacli  of  w 


Gem 
both  tei' 
ratio  «4 


If  ^i 
in,!?  that 
the  ([uan 
iished  by 


155. 

more  ra 

Since 
least  four 

Bif. 

are  calle 

If  a: 

portion  w 


PROPOHTION. 


1133 


if  it  is  less  tlian  1,  and  diininishod  if  it  is  greater  than  1 ; 
that  is,  it  will  be  l)r()Ujj;lit  ncaivr  to  unity. 

ExAMiM.K.     lii't  tlioorigiiml  ratio  bo  2  :  5  =  ^     If  wo  rt'|)out«'dly  add 
1  to  both  nuiiu'rator  and  dcinoiiiiiiator  of  tho  fraction,  we  hhall  huve  tho 

Hi'iicH  of  IVactioiiH, 

-    !'     '    "    etc 

cMcli  of  which  ia  greater  than  tho  preceding.  becuuBO 


K  =  ft"(T ;       whence,     %  >  |. 
"  •       whence,     i 


7  ~  it  —  4a  • 


etc. 


whence,     ^ 


etc. 


General  Proof.  Ix't  a  :  b  l)e  the  original  ratio,  and  lot 
Iiotii  tonus  be  increased  by  tho  quantity  n,  luaiunj;  the  new 
ratio  «-f  «  :  b-^u.     The  now  ratio  minus  tho  old  one  will  be 

{b  —  a)  u 
>  4-  bu' 

If  b  is  greater  than  a,  this  f(nantity  will  be  positive,  show- 
ing,' tliat  the  ratio  is  increased  by  adding  v  If />  is  less  than  ^/, 
tho  ([uantity  will  bo  negative,  showing  that  the  ratio  is  diniin- 
isliod  by  adding  u. 

♦■■•-♦ 


CHAPTER    II. 

PROPORTION. 

155.  Def.    Proportion  is  an  equality  of  two  or 
more  ratios. 

Since  each  ratio  has  two  terms,  a  proportion  must  have  at 
least  four  terms. 

Def.    The  terms  which  enter  into  two  equal  ratios 
are  called  Terms  of  the  proportion. 

If  rt  :  J  be  one  of  the  ratios,  and  p  :  q  the  other,  the  pro- 
portion will  be, 

a  :  b  ^=.  2) '.  q.  (1) 


134 


PROPORTION. 


I. 


I** 


A  proportion  is  sometimes  written, 

a  :  h    : :    p  :  q, 

which  is  road,  "  As  a  is  to  b  ho' is  jt  to  5."  The  first  form  is  to  be  pre. 
Icrred,  because  no  other  slg'n  than  that  of  equality  is  necessary,  but  the 
ecjuation  may  be  read,  "  As  a  is  to  6  so  is  p  to  g,"  whenever  that  expres 
Bion  is  the  clearer. 

Def.  The  first  and  fourth  terms  of  a  proportion  arc 
called  the  Extremes,  the  second  and  third  are  called 
the  Meeins. 

Theorems  of  Proportion. 

150.  Theorem  I.  In  a  proportion  the  product  of 
the  extremes  is  equal  to  the  product  of  the  means. 

Proof.  Let  us  write  the  ratios  in  the  proportion  (1)  in  tlic 
form  of  fractions.     It  will  give  the  equation, 

a  _'p 
b~q 


(3) 


Multiplying  both  sides  of  this  equation  by  Iq,  we  shall  have 

aq  =  bp.  (3) 

Cor.  If  there  are  two  unknown  terms  in  a  propoi-- 
tion,  they  may  be  expressed  by  a  single  unknown 
symbol. 

Example.    If  it  be  required  that  one  quantity  shall  be  to 

another  as  p  to  q,  we  may  call  the  first  px  and  the  second  qx, 

because 

2)X  '.  qx  z=z  p  \  q  (identically). 

157.  Theorem  II.  If  the  means  in  a  proportion  be 
interchanged,  the  proportion  will  still  be  true. 

Proof.  Divide  the  equation  (3)  by  pq.  We  shall  then 
have,  instead  of  the  proportion  (1), 

or  a  '.  p  ^=  b  :  q. 


De, 

changi 
2)ortioj 

The 
and  the 


158 

or  dim 
conseq 
still  be 

EXAA 

the  u,.iiec( 
each  ante 

Diniii 
become, 

Increi 

These 
Gene 


If  W( 

side,  it  \ 


that  18, 
IntH 


FBOPOUTION. 


135 


Def.  The  proportion  in  wliicli  tlie  means  are  inter- 
changed is  called  the  Alternate  of  the  original  pro- 
portion. 

The  following  examples  of  alternate  proportions  should  be  studied, 
and  the  truth  of  the  equations  proved  by  calculation  : 

1:2=    4:8,       alternate,     1:4    =2:8. 
2:3=6:9;  "  2  :  G     =  ;3  :  <». 

5  :  2  =  25  :  10;  "  5  :  25  =  2  :  10. 

158.  Theorem  HI.  If,  in  a  proportion,  we  increase 
or  diminish  each  antecedent  by  its  consequent,  or  each 
consequent  "by  its  own  antecedent,  the  proportion  will 
still  be  true. 

Example.    In  the  proportion, 

5  :  2  =  25  :  10, 

the  i^.itecedents  are  5  and  35,  the  consequents  2  and  10  (^  148).     Increasing 
each  antecedent  by  its  own  consequent,  the  proportion  will  be 

5  +  2  :  2  =  25  +  10  :  10,        or        7  :  2  =  35  :  10. 

Diminishing  each  antecedent  by  its  consequent,  the  proportion  will 
become, 

5  -  2  :  2  =  25  -  10  :  10,        or        3  :  2  =  15  :  10. 

Increasing  each  consequent  by  its  antecedent,  the  proportion  will  be 

5  :  2  +  5  -  25  :  10  +  25,        or        5  :  7  =  25  :  35. 

These  equations  are  all  to  be  proved  numerically. 

General  Proof,     Let  us  put  the  proportion  in  the  form 

b        q  ^  ^ 

If  we  add  1  to  each  side  of  this  equation  and  reduce  each 
Bide,  it  will  give 

a  -{■  b  _  p  -}-  q  ^ 

h      ^      q     ' 
that  is,  a  -\-  h  :  h  =  p  -{-  q  :  q,  (5) 

In  the  same  way,  by  subtracting  1  from  each  side,  it  will  be 
a  —  b  :  b  =  p  —  q  :  q.  (6) 


'  <i 


[  ■»' 


1' 


(. 


138 


proportion: 


If  we  invert  the  fractions  in  equation  (4),  the  latter  will 
become 

a      2J 

By  adding  or  subtracting  1  from  each  side  of  this  equation, 

and  then  again  inverting  the  terms  of  the  reduced  fractions, 

we  shall  lind, 

a  :  h  -{-  a  =  p  :  q  -h  P', 

a  :  b  —  a  =  p  :  q  —p- 

The  form  (5)  was  formerly  designated  as  form<^d  "  by  composition,"' 
and  (6)  as  foniied  "  by  division."  But  these  tenns  are  now  useless,  be- 
C'iuse  all  the  above  forms  are  only  special  cases  of  a  more  general  one  to 
be'  now  explained. 

151).  Theoiem  IV.  If  four  quantities  foi-m  the  pro- 
portion 

a  :  I  =  c  :  d,  (a) 

and  if  m,  n^  ^?,  and  q  be  any  multipliers  whatever,  we 
shall  have 

ma  +  nb  :  ^?a  -^  qh  —  mc  -\-  nd  :  'pc  +  qd. 
Proof.     The  projiortion  {a)  gives  the  equation, 

a  _  c 

b~'d 

p 
Multiplymg   this  equation   by  -  and  adding   1   to  each 

member, 

^  +  1  -  ^-  4-  1 
qb^^-  qd^    ' 

Reducing  each  member  to  a  fraction  and  inverting  the 

terms, 

qb  qd 

pa  -{-  qb       pc  -\-  qd 
Dividing  both  members  by  q, 

-J—  =  -A (7) 

2M  +  qb       jw  -\-  q^^ 

The  original  proportion  {a)  also  gives,  by  inversion. 


PROPORTION. 


137 


h 

a 


d 


from  which  we  obtain,  by  multiplying  by  ^,  adding  1,  etc., 


P 

qb  -\-  pa  _  qd  +  pc 

pa       ~       pc      ' 

a       c 

pa  -\-  qb~  j)c  +  qd 
(8)  X  m  +  (7)  X  w  gives  the  equation, 

ma  +  nh  _  mc  -\-  nd 
pa  -{■  qh  ~  2>c  +  qd  * 
or  ma  +  nh  :  pa  ^  qh  —  mc  +  nd  :  pc  +  qd, 

which  is  the  result  to  be  demonstrated. 


(8) 


(9) 


160.  Theorem  V.  If  each  term  of  a  proportion  be 
raised  to  the  same  power,  the  proportion  will  still 
subsist. 


Proof.     If 


a 


or 


a 
b 


V 

V 

q 


then,  by  multiplying  each  member  by  itself  repeatedly,  we 
shall  have 

«2    rpl  ^ 


¥ 


72  » 


a^  _  i>^ 

T3' 


Hence,  in  general. 


etc. 


etc. 


a'^ 


q^. 


Cor. 
then 
and 


If 


in  —  p 

a  :  b  =1  p  •  q, 
a^  :  a^  ±b"'  =  p^  :  p^  ±q^', 
a^  ±b^  :  b^  =  p^  ±  ^'*  :  q^. 

Theorem  VI.  When  three  terms  of  a  proportion 
are  given,  the  fourth  can  always  be  found  from  the 
theorem  that  the  product  of  the  means  is  equal  to  that 
of  the  extremes. 


i 


I.  I 


ii 


1^8 


PROPORTION. 


"'^/"e  have  shown  that  whenever 

a  '.  b  z=  p  :  q^ 
then  aq  ■=  bp. 

Considering  the  different  terms  in  succession  as  unknown 
([nuntities,  we  find, 

hp 

1  =  ^-^1, 
P 


V  = 


-7  = 


aq 

'b' 

bp 

a 


Cor,  1.     If,   in  the  general  equation    of  the   first 

degree 

ax  +  hy  —  c, 

the  terfn  c  vanislies,  the  equation  detennines  tlie  ratio 
of  the  unknown  quantities. 

ax  -{-  by  =:  0, 

ax  =  —  bi/, 

X  _       b 

y~~a' 

X  \  y  =.  —  b  \  a. 

Cor.  2.  Conversely,  if  the  ratio  of  two  unknown 
quantities  is  given,  the  relation  between  them  may  be 
expressed  by  an  equation  of  the  first  degree. 


Proof.     If 
then 

and 

or 


The  Mean  Proportional. 

KM.  Def.  When  the  middle  temis  of  a  proportion 
are  equal,  either  of  them  is  called  the  Mean  Propor- 
tional between  the  extrenn^s. 

The  fact  that  b  is  the  mean  proportional  between  a  and  c 
is  eXj^^ressed  in  the  form, 

a  '.  b  :=^  b  '.  c. 


PROPORTION. 


139 


Theorem  I  then  gives,  b^  =  ac. 

Extracting  the  square  root  of  both  members,  we  have 

Hence, 

Theorem  VII.  The  mean  proportional  of  two  quan- 
tities is  equal  to  the  square  root  of  tlieir  product. 

Multiple  Proportions. 

163.  We  may  have  any  number  of  ratios  equal  to  each 

-^ther,  as 

a  \  1)  ■=  c  '.  (I  ■=.  c  \  f,  etc. 

G  :  4  =  9  :  G  :=  3  :  2  =1  21  :  14.  {a) 

Sucli  proportions  are  sometimes  written  in  tlie  form 

G  :  9  :  3  :  21  '^  4  :  G  :  2  :  11.  (i) 

In  the  form  {b)  the  antecedents  are  all  written  on  one  side 
of  the  equation,  and  the  consequents  on  the  other.  Any  two 
numbers  on  one  side  then  have  the  same  ratio  as  tlie  cor- 
re>^pondinij;  two  on  the  other,  and  the  proportions  expressed  by 
this  equality  of  ratios  are  the  alternates  of  the  original  propor- 
tions («).     For  instance,  in  the  proportion  {V)  we  have, 

G,  which  is  the  alternate  of  G 

2,      ''        "  "  G 

14,     "        "  "  G 

14,    "         ''  "  9 


6 

:  9 

4 

6 

:  3 

4 

6 

:  21 

— 

4 

9 

:  21 

— 

6 

4 

— 

9  : 

G. 

4 

— 

3  : 

2. 

4 

21  : 

14. 

G 

21  : 

14. 

,'^i  ■; 


I  i       •! 


163.  A  multiple  proportion  may  also  be  expressed  by  a 
number  of  equations  equal  to  that  of  the  ratios.    Since 

a  '.  h  =.  c  \  d  ^=^  G  '.  f,  etc., 

let  us  call  r  the  common  value  of  these  ratios,  so  that 

a 


1  =  '' 


c 

d 


etc. 


Then 


a  =  rb, 

c  =  rd, 
e  -  rf, 


{c) 


140 


PROPORTION. 


will  express  the  same  relations  between  the  quantities  a,  h,  c, 
d,  e,  f,  etc.,  that  is  expressed  by 

a  :  b  =  c  :  d  =  e  :  f,  etc.,  {a) 

or  a  :  c  :  e  :  etc.  =  b  :  d  :  f  :  etc.  [(/) 

It  will  be  seen  that  where  r  enters  in  tlie  form  (c)  there  is  one  more 
equation  than  in  the  first  form  (a).  [In  this  lorui  each  =  represents  sm 
eciuatiou.]  This  is  because  the  additional  quantity  r  is  introduced,  by 
eliminating  which  we  diminish  the  nuniber  of  equations  by  one,  as  iu 
eliminating;  an  unknown  quantity. 

164.  Tlieorem.  In  a  multiple  proportion,  the  sum 
of  any  number  of  the  antecedents  is  to  the  sum  of  tlie 
corresponding  consequents  as  any  one  antecedent  is  to 
its  consequent. 

Ex.     We  have  ^  —  zr:L=  .^—  -^'   Then 


„,    ,         2       6       10      12 
Wehave-  =  ^=^^  =  -- 


30 


2  +  6  +  10  +  12 
5  +  15  +  25  +  30 
which  has  the  same  value  as  the  other  four  functions. 


75' 


General  Proof.     Let  ^,  B,  C,  etc.,  be  the  antecedents,  and 
«,  hy  c,  etc.,  the  corresponding  consequents,  so  that 

A  :  a  =  B  :  b  =  C  :  c,  etc.  (1) 

Let  us  call  r  the  common  ratio  A  :  a,  B  :  b,  etc.,  so  that 

A  =  ra, 

B  =  rb, 

C  =  re. 

etc.    etc. 

Adding  these  equations,  we  have 

A  -\-  B  +  C  -}-  etc.  z=zr{a-\-b-{-c-\-  etc.), 

A  +  B  -\-  C  4-  etc. 


or 


=  r; 


a  -\-  b  -\-  c  -\-  etc. 

that  is,  the  ratio  A -{- B -\- C -\- etc.  :  a -\- b  +  c -\- etc.  is  equal  to 
r,  the  common  value  of  the  ratios  A  :  a,  B  :  b,  etc. 


W    ' 


PROBLEMS 


I.  A  map  of  a  country  is  made  on  a  scale  of  5  miles  to 
3  inches. 


PROPOUTION. 


141 


(1.)  What  will  be  the  length  of  8,  12,  17,  20,  33  miles  on 
the  map? 

(2.)  How  many  miles  will  be  represented  by  (I,  8,  IG,  20, 
21)  inches  on  the  map  'i 

Rem.  1.  If  X,  y,  z,  u,  v  be  the  required  spaces  ou  the  map,  we  sliall 
have 

5  :  S  =  8  :  X  =  12  :  y,  etc. 

If  a,  b,  c,  etc.,  be  the  required  number  of  miies,  we  shall  have 

3  :  5  =  6  :  a  =  8  :  6  -  16  :  r,  etc. 

Rem.  2.  When  there  are  several  ratios  compared,  as  in  this  problem, 
it  will  be  more  convenient  to  take  the  inverse  of  the  common  ratio,  and 
multiply  the  antecedent  of  each  following  ratio  by  it  to  obtain  the  conse- 
quent.    In  the  first  of  the  above  proportions  the  inverse  ratio  is  g,  and 

cc  =  ^  of  8,    y  =  §  of  13,  etc. 
In  the  second,       a  =  f  of  6,    6  =  |  of  8,  etc. 

2.  To  divide  a  ^iven  quantity  A  into  three  parts  which 
shall  be  proportional  to  the  given  quantities  a,  b,  c,  that  is, 
into  the  parts  x,  y,  and  z,  such  that 

X  '.  a  z=i  y  '.  h  =^  z  :  c, 
or  X  :  y  :  z  =1  a  :  h  \  c. 

Solution.    By  Theorem  IV, 

X y  z  _x  -\-  y  -^^  z  A 

a       b        c       a  -\-  b  ■\-  c       a  -\-  b  ^  c 
Therefore, 

aA  _        ^^  _        ^^ 


♦  ■ 


X 


3.  Divide  102  into  three  parts  which  shall  be  proportional 
to  the  numbers  2,  4,  11 

4.  Divide  1000  into  five  parts  which  shall  be  proportional 
to  the  numbers  1,  2,  3,  4,  5. 

5.  Find  two  fractions  whose  ratio  shall  be  that  of  a  :b,  and 
whose  sum  shall  be  1. 

6.  What  two  numbers  are  those  whose  ratio  is  that  of  7  :  3 
and  whoso  difference  is  24. 

7.  What  two  numbers  are  those  whose  ratio  is  m  :  71,  and 
whose  difference  is  unity  ? 

8.  Find  x  and  y  from  the  conditions, 

X  :  y  =z  a  :  bj 
ax  —  by  =  a  +  b. 


. } 


r 


14:^; 


PllOPOUTION. 


9.  Show  that  if        a  :  b  —  A  \  By 

c  :  d  =  C  \  D, 
we  must  also  have      ac  :  bd  =  AC  :  BD. 


10. 


Ilav 


ing  given  x  —  ay,  find  the  value  of 


II.  Having  dven 


to  to' 


X  +  %y 


-'y 


X  — 


fiud  the  value  of 


y 


X 


12.     If 


a 


h  =  p  \  q, 


prove 


and 


d^  +  b^ 


CfP-  +  />" 


ft-' 


p^ 


a  +  /> 


rt 


re+l 


=  />^  +  //''  :    -  — ■ 


«  4-  (^ 


2)n  _j..  ^? 


13- 


Tf 


r?  +  /^;  4-  <•  +  d 


a 


b  +  c  —  d 


a  +  b 


d 


show  that 


« 


a  —  b  —  c  +  d'' 

e  :  d. 


14.  A  yenr's  profits  were  divided  among  three  partners,  A, 
B,  Mild  C,  pro})(>rtional  to  the  numbers  'i,    3,  and  7.     If  L 


num 
woul 


should  pay  B  $1250,  their  shares  would  be  equal.     What  w 
he  amount  divided  ':* 


ras 


Ii 


15.  in  a  fii'st  year's  partnership  between  A  and  B,  A  had 
2  shares  and  B  had  5.  In  the*  second  year,  A  liad  3  and  B  had  4. 
In  the  second  year,  A's  profits  were  ^320('  greater  and  B's  were 
$1700  greater  than  they  were  the  first.  What  was  each  year's 
profits!^ 

16.  In  a  poultiy  yard  there  are  7  chickens  to  every  2  ducks, 
and  3  due  s  to  every  2  geese.  How  many  geese  were  there  to 
every  42  chickens? 

17.  A  drover  started  with  a  herd  containing  4  horses  to 
every  9  cattle.  He  sold  148  horses  and  108  cattle,  and  then 
had  1  horse  to  every  3  cattle.  How  many  horses  and  cattle 
had  he  at  first  ? 

18.  If  a  bowl  of  punch  contains  a  parts  of  water  and  b 
parts  of  wine,  what  is  the  ratio  of  the  wine  to  the  wliole 
punch  ?  What  is  the  ratio  of  the  water  ?  What  are  the  sums 
of  these  ratios  ? 


PROPORTION. 


143 


19.  One  ingot  consists  of  equal  parts  of  gold  and  silver, 
wliile  another  has  two  parts  of  gold  to  one  of  silver.  If  I 
combine  e([ual  weights  from  these  ingots,  what  proportion  of 
the  compound  will  he  gold  and  what  [)roportion  silver';* 

20.  What  will  he  the  proportions  if.  in  the  preeeding  proh- 
leni,  I  combine  one  ounce  from  the  Wv»t  ir;got  with  three  from 
tiic  second? 

21.  One  cask  contains  a  gallons  of  water  and  h  gallons  of 
alcohol,  while  another  contains  m,  gallons  of  water  and  u  of 
alcohol.  If  I  draw  one  gallon  from  each  cask  and  mix  them, 
what  will  be  the  ([uantities  of  rdcohol  and  water  r" 

22.  What  will  be  the  ratio  of  the  JKpiors  in  the  last  case,  if 
I  mijx  two  parts  from  the  first  cusk  with  one  from  the  second  ? 

23.  What  will  it  be  if  I  mix  p  parts  from  the  first  with  // 
parts  from  the  second  ? 

24.  A  goldsmith  has  two  ingots,  each  consisting  of  an  alloy 
of  gold  and  silver.  If  he  combines  two  parts  from  the  first 
ingot  with  one  from  the  sec(md,  he  will  have  e(|ual  parts  of 
gold  and  silver.  If  he  combines  one  ])art  from  the  fir«t  with 
two  from  the  second,  he  will  have  3  parts  of  gold  to  5  of  silver. 
AVhat  is  the  composition  of  each  ingot  ? 

SuGGESTiox.  Call  r  the  ratio  of  the  weight  of  gold  in  t'ne  first  ingot 
to  the  whole  weight  of  the  ingot ;  then  1—7'  will  be  the  ratio  of  tlie  sil- 
ver in  the  first  to  the  whole  weight  of  the  ingot.  See  the  following 
question. 

Note.  Problems  18-24  f^r™  a  graduated  series,  introductory  to  the 
processes  of  Problem  24. 

25.  Point  out  the  mistake  which  would  be  made  if  the 
solution  of  the  preceding  problem  were  commenced  in  the  fol- 
lowing way  : 

If  the  first  ingot  contains  p  parts  of  gold  to  q  parts  of  silver,  and  the 
second  contains  r  parts  of  gold  to  a  of  silver,  then 

Two  parts  from  the  first  ingot  will  have  2p  of  gold  and  2q  of  silver. 

One  part  from  the  second  ingot  will  have  r  of  gold  and  s  of  silver. 

Therefore,  the  combination  will  contain  2p  +  r  parts  of  gold,  and 
'iq  +  s  parts  of  silver. 

Show  also  that  if  we  subject  p,  q,  r,  and  s  to  the  condition 

the  process  would  be  correct. 

-  26.  Show  that  if  the  second  term  of  a  proportion  bo  a 
mean  proportional  between  the  third  and  fourth,  the  third 
will  be  a  mean  proportional  between  the  first  and  second. 


f 


,,  Mfl 


BOOK    V. 
OF    POWERS    AND    ROOTS. 


CHAPTER     I. 

INVOLUTION. 


Case  J.    Involution  of  Products  and  Quotients. 

UJ5.  Def.  Tlie  result  of  taking  a  quantity,  A, 
n  times  ati  a  factor  is  called  the  i/*'*  power  of  vl,  and 
as  already  known  may  be  written  either 

AAA^  etc.,  n  times,     or    A^. 

Def.  The  number  n  is  called  the  Index  of  the 
power. 

Def.  Involution  is  the  operation  of  finding  the 
powers  of  algebraic  expressions. 

The  operation  of  involution  may  always  be  expressed  by 
the  application  of  tlie  proper  exponent,  the  expression  to  be 
involved  being  inclosed  in  parentheses. 

Example.    The  n^^  power  oi  a  -\- h  \b  («  +  ^)" 
The  w"^  power  of  aJ)c  is  {abcY. 

166.  Involution  of  Products.  The  n^^  power  of  the 
product  of  several  factors  a,  &,  c,  may  be  expressed  without 
exponents  as  follows : 

ahcabcabc,   etc., 

each  factor  being  repeated  n  times. 


INVOLUTION. 


145 


Hero  thoro  will  bo  alto^'cthcr  n  a%  n  b\,  and  n  c's,  so 
that,  using  exponents,  the  whole  power  will  be  a^b'^c^^  (g  GG,  G7). 

Hence,  (abt)»'  =  a^b'^c^. 

That  is, 

Tfieoreni.     The  power  of  a  product  is  eqiuil  to  the 
product  of  the  powers  of  the  several  factors. 

107,  Invohdion  of  Quotients.    Applying  the  same  mcthotls 

to  fractions,  we  find  that  the  w^''  power  of  -  is  —  •    For 

y     T 


/xV^      X  X  X      ^ 

\)  =  ~  — ,  etc.,  n  times; 

^y'      y  y  y 


XXX,  etc.,  n  times 
yyy,  etc.,  n  times 


(§  109) ; 


tin 
o 


EXERCISES 

Express  the  cubes  of 

,  ab 

I.     abc.  2.     — • 

c 

mn  a  4-  b 

pq  ^      a~b 

Express  the  n*f^  powers  of  the  same  quantities,  the  quanti- 
ties between  parentheses  beinnr  treated  as  siiigr.,  symbols. 


3.     abc'"^. 

,      mn  {a  -f-  b) 


Case  II.    Involution  of  Powers. 

1G8.  Pkoblem.     It  is  required  to  raise  the  quantity  a^  to 
the  n*^  power. 

Solution.     The  n*^  power  of  a"'  is,  by  definition, 

a"i  X  n"^  X  a%  etc.,  n  times. 

By  §  66,  the  exponents  of  a  are  all  to  be  added,  and  as  the 
exponent  m  is  repeated  n  times,  the  sum 

m  -\-  7n  -{■  m  i-  etc.,  n  times, 
is  mn.     Hence  the  result  is  a^^,  or,  in  the  language  of  Algebra, 

10 


V 


■  \ 


•\ 


146 


INVOLUTION. 


Ilcncc, 

Thmrom.  If  any  power  of  a  (iiiantity  is  itself  to  Ijo 
raised  to  a  i)ower,  the  indices  (  f  the  powers  must  be 
multiplied  together. 

EXAMPLES. 

Note.     Tt  will  l)o  pocn  tliat  tins  thof)n'ni  coinoidoa  with  thnt  of  Cawe  I 
when  liny  of  thi;  factors  havu  tlu)  cxpoiii'iit  unity  uniJeratoocl. 

EXERCISES. 


Write  the 

cubes  < 

:)f  the  following  (iiuintities: 

1.     3./y. 

4a 
2.     y                      3. 

r/"». 

4.     bx^. 

5.     ^>^/.2^/<".                  6. 

Write  the 

n^^  powers  of 

7.     a. 

8.      ^/2/>.                         9. 

aWc. 

10.     lO^a'K 

II.     2^>%/.                 12. 

{n^h){c^-(l) 

13-     (•<•  +  2/)  (»•  -  .'/)• 

14.     7  (rt  +  <!»  —  c)  (a  —  />V'- 

/1/2.S'.  7«  (rt  +  i  —  cY  {(t  —  ?>)"''• 


a 

h 

ah  {c  -  (If 


15.  ;•  16. 

18. 

'9-  (^a-h)c^ 
Reduce: 

20.  {'lami^y. 

22.  2a{—'il^imi^Y. 

24.  (rtZ*")*. 


a* 


17. 


^  + «/ 


^«s. 


X  —  y 
m^a^b^» 


^ny2H 


21.     (—Ihanx^)^. 

23.     {7pq'^r^)\ 

25.     (2«2r3)'i.  26.     (»?")". 

Note  1.  If  the  student  find  any  of  these  exponential  expressiona 
difficult  of  expression,  he  may  first  express  them  by  writing  each  quantity 
a  number  of  times  indicated  by  its  exponent. 

Note  2.  The  student  is  expected  to  treat  the  quantities  in  paren- 
theses as  single  symbols. 


lyvoLurioy. 


147 


Hkm.  The  pi'occdiii^'  tlu'oiviii  liiids  w  practical  a|>}>licati()U 
when  it  is  nccestiary  to  raiso  ii  small  iiuiiiIut  to  a  lii^^'h  power. 
If,  lor  cxaniplc,  we  luivc  to  raise  'i  to  the  ;iOih  power,  wo 
fihoiilil,  without  this  theorem,  have  to  multiply  by  '4.  no  le^S 
than  v'.J  times,  lint  we  may  also  proeeeil  thus: 
t^   =  4, 

2<  =  a^-5i3  =  4.4      =  in, 

216  _  28.08   _  o,-,(ja       _  oSolJO, 
2?!  ~  ^m.-i^  -  2i«.:>r)G  =  lOTTT'^'lf;, 

280  _  2*«.;efl  =  :>24.G4  =  i():;jr4i8-.>4. 


Case  of  Noj^jitivo  Exponents. 

109.  Tlu!  })rec'e(lin^-  theorem  may  be  applied  to  negative 
exponents.      My  the  delinition  of  such  exi)onents, 


aP 

-,-  =  aPlni. 


(1) 


liaising  the  lirst  member  to  the  n^^  i)0wcr,  we  have, 

(nvY      a»P  , 

\b'i  I       Im 

This  is  tho  same  result  wc  should  get  by  ai)plying  I  ho 
theorem  to  the  second  member  of  (1),  tmd  proves  the  proposi. 
tion. 

EXERCISES. 

Express  the  Gth  powers  of 


I.     alr^. 

2. 

aVj-\ 

3.     nmir^. 

4. 

a~»'  b-'K 

5.     {(iJ^hY{a-l 

>)-'- 

6. 

{x  4-  !/Y  {•>'  +  2)-*. 

a-P 

0 

{a  +  /;)-"' 

7-     W 

8. 

(a  —  b)'"' 

Reduce : 

9.     [{a  +  h)-^  {a  - 

-  ^)Y- 

10. 

{((b-'c-^y. 

ir.     («J-i6'-2)-5. 

12. 

{mhrJyi. 

13.    {x^iryi. 

14. 

(a^hH'  ^y. 

After  forming  the  expressions,  write  them  all  with  positive 
ex]-)onents,  in  the  form  of  fractions. 


148 


INVOLUTION. 


I 


Algebraic  Signs  of  Powers. 

170.  Since  the  continued  product  of  any  number  of  posi- 
tive factors  is  positive,  all  the  powers  of  a  positive  quantity  are 
positive. 

By  §  20,  the  product  of  an  odd  number  of  negative  fac- 
tors is  negative,  and  the  product  of  an  even  number  is  positive. 
Hence, 

TJieorem.  Tlie  even  powers  of  negative  quantities 
are  positive,  and  the  odd  powers  are  negative. 

EXAMPLES. 

(—  ((f  =  a^;     {—  a)^  =  —  a^\     (—  ay  =  a*,    etc. 


i 


Find  the  value  of 
1.      (-2)2. 

4.   i-oy. 

7.     {-n-hy. 


lO. 


i-a) 
13.     (-1) 


'an 
2?i 


EXERCISES, 


2.       (-3)3. 

5.     (-5)^ 

8.    {—v}?iy. 

2n- 1 
2n+l 


11. 


(-/.) 

14.      (-1) 


3-     4^. 

6.     {-by. 

9-    i-pqy- 

{-a-h) 


12. 


1n-\ 


15-     (-1) 


2n-\ 


Case  III.  Iiivolutiou  of  Binomials — tlie  Bino- 
mial Theorem, 

Ivi.  It  is  required  to  find  the  7//^  power  of  a  binomial. 
1.  Let  a  +  i  be  the  binomial ;  its  w^  power  may  be  written 

(«  +  ly. 

Let  us  now  transform  this  expression  by  dividing  it  by  (0\ 
and  then  multiplying  it  by  «",  which  will  reduce  it  to  its  orig- 
inal value.     We  have  (§  1G7), 

{(I  +  hy       (a  4-  hy      L    ,   by 

Multiplying  this  last  ex])rcssion  by  ««,  by  writing  this 
power  outside  the  parentheses,  it  becomes 


a" 


(^  ^  T 


(1) 


INVOLUTION. 


149 


which  is  equal  to  {a  -f  h)'K     Next  let  us  put  for  shortness  x  to 
represent  - ,  when  the  expression  will  become 

{a  +  l)^  =  cO^  (1  +  xyK  (2) 

2.  Now  let  us  form  the  successive  powers  of  (1  +  ^y.     We 
multiply  according  to  the  method  of  §  70: 


Multiplier, 


(1  j^xy  =  l+x 

I  +x 


Multiplier, 


l-\-x 

+  X    +  :?-2 

(1  +  xf 

—  1  +  o,,.  _^  .,.i 

1  +  a; 

1  +  2a;  4-  a;2 

X  +  'Ix'i  +  x^ 

(1  4-  ^Y 

—  1  4.  3.^  +  3a;2  -^  7^ 

l  +  .r 

1  4-  3a;  +  3a-2  -f  a;3 

a;  +  3a;2  +  3^-3  -J 

Multiplier, 


(1  +  xf  r=  1  +  42;  4-  G^'-J  4-  4.^-3  +  a.-4 

It  will  be  seen  that  whenever  we  multiply  one  of  these 
powers  by  1  -\-  x,  the  coefficients  of  x,  x\  etc.,  which  we  add 
to  form  the  next  higher  power  are  the  same  as  those  of  the 
given  power,  only  those  in  the  lower  line  go  one  place  toward 
ilie  right.  Thus,  to  form  (1  4-  a-)*,  we  took  the  coefficients  of 
(I  4-  xY^  and  wrote  and  added  them  thus  : 

Cocf.  of  (1  +  xY,         1,    3,     3,     1. 

1,     3,    3,     1. 

Coef.  of  (1  4-  xY,        1,     4,     6,     4,     1. 

It  is  not  necepsary  to  writo  the  numbers  under  each  other  to  add 
them  in  tliis  way  ;  wc  hav(>  only  to  add  eac!'  number  to  the  one  on  tlie 
left  in  tlie  same  line  to  form  the  eorrespondin^^  nunibtTof  the  line  hiOcrw. 
TluiH  we  can  form  tlu;  coellicieuts  of  the  successive  powers  of  .v  at  si<j:lit 
as  follows:  The  first  figure  in  each  lino  is  1  ;  the  next  its  the  coefficieut 
of  X ;  the  third  the  coefficient  of  x^,  etc. 


.  I 


loO 


INVOLUTION. 


V 


% 

If 


First  power,    n  -—  1,    coefficients,  1,  1. 


Second 

a 

n  = 

^, 

'Hiird 

a 

n  — 

3, 

Fourth 

a 

n  — 

'1, 

Fifth 

li 

n  =^ 

5, 

Sixtii 

a 

n  — 

0, 

etc. 

etc. 

« 

1,  2, 

1. 

tt 

1,  3, 

3,     1. 

ft 

1,  4, 

0,     4, 

1. 

tt 

1,    0, 

10,  10, 

5,  1. 

u 

1,  (J, 

15,  20, 
etc. 

15,  6,  1 

It  is  evident  that  tlie  first  quantity  is  always  1,  and  that 
the  next  coefficient  in  each  line,  or  the  coefficient  of  x,  is  n. 
The  third  is  not  evident,  but  is  really  equal  to 

^^.  (*) 

as  will  be  readily  found  by  trial;  because,  beginning  with 

71  =  3, 

3.2        ,       4-3         ,^       5.4 


The  fourth  number  on  each  line  is 

n  (n  —  1)  {)i  —  2) 
2.3 
Thus,  beginning  as  before  with  the  third  line,  wiiere  n  =  3, 


2.3  '  "2.3  ' 


10  =  - 


.4.3 

2-3" 


etc. 


(^•) 


3.  These  several  quantities  are  called  Binomial  Coeffi- 
cients. In  Avriting  them,  we  m^y  multiply  all  the  denomi- 
nators l)y  the  liictor  1  witiiout  changing  them,  so  that  there 
will  be  as  many  factors  in  the  denominator  as  in  the  numerator. 
The  fourtii  column  of  coefficients,  or  {c),  will  then  be  written, 

3.2.1  4.3.2  5.4.3 


1.2.3'        r2.3'         1.2-3 


etc. 


4.  We  Cim  llnd  all  the  l)inoniial  coefficients  of  any  jiowcr 
when  we  know  the  yaluc  of  u. 

The  numerator  and  denominator  of  the  second  coefficient 
will  contain  two  factors,  as  in  (h)  ;  of  the  tlvir<l,  three  factors, 
as  in  {(•) ;  and  of  the  .s'^,  .v  factors,  whatever  <«?  may  be. 

In  any  coefficient,  the  first  factor  in  the  numerator  is  n, 
the  second  ii  —  \,  etc.,  each  factor  ])eing  less  by  unity  than  the 


fr 


INVOLUTION. 


151 


preceding  one,  until  we  come  to  the  s^'*  or  last,  which  will  be 
n  —  s  +  1. 

Such  a  i)roduct  is  written, 

u  (n  —  1)  {n  —  2)....  {)i  —  s  -{-  1). 

The  dots  .stand  for  any  nnniber  oi"  omitted  factors,  because 
6^  may  be  any  number.  \Vc  have  written  4  of  the  s  factors,  so 
flijit  .§  —  4  arc  left  to  be  represented  by  the  dots. 

Tire  denominator  of  the  fraction  is  the  product  of  the  s 

factors, 

1.2.3  .  .  .  .  s, 

each  factor  being  greater  by  1  than  the  preceding  one,  and  tlie 
(lots  standing  for  any  number  of  omitted  factors,  according  to 
the  value  of  s.     Thus,  tlie  s^^  coefficient  in  the  7i^^  line  will  be 

11  (n  —  l){n  —  2)....  {n  —  .s  +  1) 

If  s  is  greater  than  ^n,  the  last  factors  will  cancel  some  of 
the  preceding  ones,  so  that  as  s  increases  from  yi  to  n,  the 
values  of  the  preceding  coefficients  will  be  repeated  in  the 
reverse  order.  Thus,  suppose  n  =  C.  Then,  by  cancelling 
common  factors, 

6-5.4.3  _  G.5  _ 

1.2.3.4  ~  r^  ~    ' 

6.5.4.3.2  _  C  _ 

1.2.3.4.5  ""  1  ~ 

6.5.4.3.2.1  _ 
1.2. 3. 4. 5.0  ~    • 

If  we  should  add  one  more  factor  to  the  numerator,  it 
would  be  0,  and  the  whole  coefficient  would  be  0. 

The  conclusion  we  have  reached  is  embodied  in  the  follow- 
ing equation,  which  should  be  perfectly  memorized  : 


(1  +  a;)»  =  1  +  vx  + 

+ 


n(n-l)^^  ,   n{}>-l){n-2) 


1.2 


x^-}- 


1.2-3.4 


1.2-3 


1     ,L,.'« 


■     I 


152 


INVOLUTION. 


EXERCISES. 

1.  Compato  from  the  formula  {d)  all  the  binomial  coefti- 
cients  for  n  =  G,  and  from  them  express  the  development  ol 

(1  +  :<f. 

2.  Do  the  same  thing  for  7i  =  8,  and  for  n  =  10. 

17*^i.  To  find  the  development  of  {a  +  i)",  we  replace  x 
by     ,  and  then  multiply  each  term  by  «". 

[See  equations  (1)  and  (2).]     We  thus  have 

{a  +  Z»)"  =  a»  +  na^'-^b  +  -^.— r—  a»-^ifi  -f  etc.  to  i« 

The  term?  of  the  development  are  subject  to  the  following 
rules : 

I.  Tlie  exponents  of  b,  or  the  second  term  of  the  hlno- 
Diial,  are  0,  1,  2,  etc.,  to  n. 

Because  U^  is  simply  1,  a"  is  the  same  as  a»b'^. 

II.  The  sum  of  the  exponents  of  a  and  b  is  ft  in  each 
term.    Hence  the  exponents  of  a  are 

91,    n  —  1,     n  —  2,    etc.,  to    0. 

III.  Tlie  coefficient  of  the  first  term  is  unity,  and  of 
the  second  n,  the  indero  of  the  power.  Each  foUoirin^ 
coefficient  may  be  found  from  the  next  preceding  one  by 
multiplying  by  the  successive  factors, 


n 


n 


2 


3 


w  -  3 

— ^-,    etc. 


IV.  If  b  or  a  is  negative,  the  sign  of  its  odd  powers 
ii'iU  be  changed,  but  that  of  its  even  powers  will  remain 
the  same. 

(Compare  §  170.)     Hence, 

'ff  (fi \\ 

{a  —  b)"^  =  «"  —  7ian-^  -] ^— - — '-  aP'-W  —  etc., 

1  •  /i 

the  terms  being  alternately  positive  and  negative. 


3.  Cc 

coctiicieu 

4.  W 

{a  +  by^ 
5.  ^v 

{"iam  +  • 

6.  W 

h^ 

7.  W 
{a  -  x) 

8.  W 

g.   W 

following 

(1 
1_ 

^■ax 

lo.   ^^ 

ments,  (l 


II 


13 


\ax 


Case 
173. 


be  any  pc 


INVOLUTION. 


153 


E  X  E  R  C  I  S  E  S  — Continued. 

3.  Compute  all  the  terms  of  {a  +  hf,  using  the  binomial 

roefficients. 

4.  What   is   the  coetMciunt   of  b^  in  the  develoi)ment  of 
[a  +  by^. 

5.  What  are  the  first  foui  terms  in  the  development  of 
{•him  +  3w)8. 

6.  What  are  the  first  three  terms  in  the  development  of 

jl  +  -I  ?    What  are  the  last  two  terms? 

7.  What  are  the  first  three  and  the  last  three  terms  of 

8.  What  is  the  development  of  la  +  -)  • 

9.  Wliat  are  tlie  first  four  terms  in  the  development  of  the 
following  binomials: 

(1  -f  X^Y  ;  (1  +  2:6-2)"  .  (1  _  ^i-^2)n  . 

10.  What  are  the  sum  and  difference  of  the  two  develop, 
ments,  (1  +  .r)^  and  (1  —  a-)^? 


Case  IV.    Square  of  a  Polynomial, 

173.    1.  Square  of  any  Polynomial.     Let 
a-\-b-\-c-\-d-\-  etc., 
be  any  polynomial.     We  may  form  its  square  thus : 

■    a  -\-  b  -\-  c  -\-  d  -{■  etc. 
(I  +  b  +  c  -\-  d  -\-  etc. 

;■'■"■  +  ab  -\-  ac  +  ad  +  etc. 

ab  i-b'^  -\-  be  +  bd  -f  etc. 

ac  -f-  be  +  c^  -f-  cd  -f-  etc. 

ad  +  bd  -\-  cd  +  d-  +  etc. 

«2  ^  ^^  4-  (;2  +  d^  4_  etc.       ~~ 
+  2ab  +  2«c  H-  2ad  +  etc. 

+  26c  -f  2bd  +  etc.  +  2cd  +  etc. 


.    ^{     1 


ii 


U^ 


■  ht 


154 


INVOLJTION. 


U 


We  thus  rciicli  tlic  following  conclusion : 

Tlwortm.  The  squja-e  of  a  polynomial  is  equal  to 
the  riiim  of  the  squares  of  all  its  terms  })lus  twice  the 
pi'odu(;t  of  eveiy  two  terms. 

y.  S(iit((rc  of  <ni  Knlirc  Fundiyn.  Sometimes  we  wish  h. 
arrange  tiie  polynomial  and  its  sfjiuiro  as  an  entire  function  ot 
some  (juantity,  for  exam[)le,  ol'./-. 

Let  us  form  the  s(j[Uiirc  of  a  -\-  hx  -\-  coi?  ■\-  dx^  -|-  etc. 

a  -\-  hx  -i-  cx^  +  dx^  +  etc. 
a  -\-  bx  ■}-  cx^  +  dj?  +  etc. 


a^  +  ahx  -\-  ncx^  +  adx^  -f  etc. 

abx  4-  ^2^-2  4-  hex-  +  hdx^  +  etc. 

acx^  +  hcx^  +   rV  +  etc. 

(uh^  4-  hdx^  +  etc. 

a^  4-  2ahx  +  (2rtc  +  V^)  x^  4-  ('^r/f/  4-  'Ibc)  0^  4-  etc. 

We  see  that : 

The  coefficient  of  x^  is  ac  -^  hh  -\-  ca. 

"  "  "  a?  is  ad  +  he  +  eh  4-  da. 

"  "  "  :c^  is  «p  4-  /yf/  4-  cc  -f  ^//>  4-  ea. 

etc.  etc. 

The  law  of  the  products  ae,  hd,  cc,  etc.,  is  that  the  first 
factor  of  each  product  is  composed  successively  of  all  the  co- 
efficients in  regular  order  up  to  that  of  the  jiower  of  a;  to  which 
the  coefficient  belongs,  while  the  second  factor  is  composed 
successively  of  the  t^ame  coefficients  in  reverse  order. 

EXERCISES. 

Form  the  squares  of 

I.     1  4- 2;f  4- ^i-^.  2.     1  4- 2:r  4- 3ir2  +  4.^8. 

3.  1  4-  •2x  4-  3.^2  4-  4.?:3  4-  5:6-5. 

4.  1  -f  2.r  4-  3a;2  4-  4.^3  4-  52-5  4-  {jx\ 
1  _  2x  4-  3.t2  _  42-3.  6.     a—h^-c  —  d. 


3rt  -\-U  —  c-\-d. 


8. 


1       , 

«  4-     —b 
a 


1 

—  t 

b 


EVd 

number 
Whe 
Whe 

Exam 

As  the 

There 
now  descr 

175. 

root  of  a^. 
li\  itself,  V 
tity  is  a^, 

Thesq 
In  the 

The  fo 

Tlieor 
])ressed  1 
dividing 

170.  . 

nal  definil 


EVOLUTION. 


irw 


CHAPTER     I  i. 

EVOLUTION    AND    FRACTIONAL    EXPONENTS. 

174:.  Dif.  The  «i"*  Root  of  a  quantity  q  is  such  a 
iminber  as,  being  raised  to  tb»*  ^''*  power,  will  produce  q. 

When  n  =  2,  the  root  is  called  the  Square  Root. 

When  71  =  3,  the  root  is  called  the  Cube  Root. 

Examples.    3  is  the  4tli  root  of  81,  because 

3.3.3-3  =  34  =  81. 
As  the  student  already  knows,  wo  use  the  notation, 

«''*  root  of  q  =  ^'q. 

There  is  another  way  of  expressing  roots  wliich  we  shall 
iKjAV  descril)e. 

175.  Division  of  Exponent fi.  Let  us  extract  the  square 
root  of  a^.  We  musi  find  such  a  quantify  as,  being  muUiplied 
hy  itself,  will  produce  a^\  It  is  evident  that  the  required  (|uau- 
tity  is  a^,  because,  by  the  rule  for  multiplication  (§§  GG,  IGG), 

a^  X  a^  =  a^ 

n 

The  square  root  of  ((^  will  be  a^,  because 

n  II  n     n 

n 

In  the  same  way,  the  cube  root  of  «"  is  a^,  because 

n  n  71. 

The  following  theorem  will  now  be  evident: 

Tlieorem.  The  square  root  of  a  power  m.ny  be  ex- 
l)ressed  by  dividing  its  exponent  by  2,  the  cube  root  by 
dividing  it  by  3,  and  the  n^^"-  root  by  dividing  it  by  n. 

170.  Fractional  Exponents.  Considering  only  the  oiigi- 
nal  definition  of  exponents,  such  an  expression  as  «'-  would 


158 


EVOLUTION. 


If 'I 


ft 


* 


w>. 


liavo  no  meaning,  because  we  cannot  write  a  1^  times.  But 
by  what  has  just  been  said,  we  see  that  a-  may  be  interpreted 
to  mean  the  square  root  of  a^,  because 

3  3- 

a^  X  a-  =  ifi. 
llonce, 

A  liactional  exponent  indicates  the  extraction  of  a 
root.  If  the  denominator  is  2,  a  square  root  is  indi- 
cated ;  if  3,  a  cube  root ;  if  n^  an  n^^  root. 

A  fractional  exponent  ;"\s  ; '.reforc  the  same  meaning  as 
the  radical  sign  V?  and  m:-;  f,c  <:^od  in  place  of  it. 

E  X  E  r.  C  I  S   :.  "■- 

Express  the  following  roots  by  exponents  only  : 

I.     V';".  2.     a/(w  +  n).  3-     -Vict  +  Hf. 

4.     ^/{a  +  bf.      5.     'Vm^  •         6.     ^a«. 

7.     v^rt^  8.     ^/{a  +  ^)«.  9.     \/(rt  4-  ^)'". 

177.  Since  the  even  powers  of  negative  quantities 
are  positive,  it  follows  that  an  even  root  of  a  positive 
quantity  may  be  either  positive  or  negative. 

This  is  expressed  by  the  double  sign  ± . 

EXERCISES. 

Expr.'ss  tlie  square  roots  and  also  the  cube  roots  and  tlie 
n^^  roots  of  the  following: 

I.     {a  +  bf.  2.     {a  +  bf.  3.     a^b. 

4.     {x  +  y)^-  5.     (^+?/)^-  6.     Cc  +  7/)\ 

178.  If  the  quantity  of  which  the  root  is  to  be  ex- 
tracted is  a  product  of  several  factors,  we  extract  the 
root  of  each  factor,  and  take  tlie  product  of  these  roots. 

Example.     The  n^'^  root  of  am^p  is  a^ni^p^,  because 
{ro'm^p^f  z=  am%  by  §§  168  and  17G. 

If  the  quantity  is  a  fraction,  we  extract  the  root  of 
both  members. 


E  VOLITION. 


157 


Proof. 


©"= 


a 


(§§1G7,  108.) 


.A 


Because  -r  taken  n  times  as  a  factor  mukcs  r,  therefore, 
ih  b 


l>v  dellnition,  it  is  the  n^''  root  of 


n 


EXERCISES. 


4a;2. 


Express  the  square  roots  of 

^'     49wi 
Express  the  cube  roots  of 
4.     27- G4. 


G4«*2^ 


2/,3 


5.     27rt 


97^/3 


Shnj/^fj 


6.     64.27«3i6. 


7.     ahh^d^. 


8. 


8^ 


m 


125icy 


» 


Express  the  n*^  roots  of 


10.     4.7. 


12. 


15. 


6)112)"' 


13- 


Ga^b^^' 


II. 


14. 


10. 


Grt2/;^ 


m 


C'«i/« 


16.  35«  «-^'*  {a  +  J)^'^  (a:  —  ?/)«  4^  (J  —  c  +  </)-'*«. 
Reduce  to  exponential  expressions : 

17.  \^a  (b  -  (y^.  18.     \/a¥c\ 
19.     '^^aPb^. 

ml  {a  4-  ft)» 


20. 


21. 


(r^  —  bf' 


Powers  of  Expressions  with  Fractional  Expo- 
nents. 

179.  Theorem,    The  j)^^  power  of  the 
equal  to  tlie  n^^  root  of  the  p-^  power. 


Ah 


\ 


S^l 


;.  ,     Sl 


158 


FJiA  CTIONA  L   L'XPOJVhWTti. 


I 


In  iilgebruic  language, 

{^/ay  =  vV. 
or  (a")"  =  (a")", 

Example.  {^/>^f  =  5i2  =  4, 

or,  in  words,  tlic  square  of  the  cube  root  of  8  (that  is,  llut 
square  of  'Z)  is  the  cube  root  of  the  square  of  8  (that  is,  of  01). 

General  Proof.    Let  us  put  x  =  tlie  ?i^^  root  of  a,  so  that 

a;«  =  n.  (1) 

Tiie  y^  power  of  this  root  x  will  then  be  xp.  (2) 

Raisijog  both  sides  of  the  equation  (1)  to  the  p^^  power,  we 

have 

x^P  z=  aP  —  pf^  power  of  a. 

The  n*^''  root  of  the  first  member  is  found  by  dividing  the 
exponent  by  w,  whieli  gives 

n^^  root  of  p^^  power  =  xp, 

the  same  expression  (2)  just  found  for  the  p^^  power  of  the 
n^^  root. 

This  theorem  leads  to  the  following  conclusion : 

1.  The  expression  ;, 

a" 

may  mean  indifferently  the  p^^  power  of  a'^,  or  the  Tith 
root  of  aJ',  these  quantities  being  identical. 

2.  The  powers  of  expressions  having  fractional  ex- 
ponents may  be  formed  by  multiplying  the  exponents 
by  the  index  of  the  power. 

EXERCISES. 

Express  the  squares,  the  cubes,  and  the  n^^  powers  of  the 
following  expressions : 


I. 

4- 


1 
aK 


2.     a^. 
5.     abK 


2 


7.      a 

9.      (' 
II.     a 

Re(lu('( 

.3.  (. 


(I 


15.     ( 

■'■(I 


REC 


6.     ab^  c^. 


180. 

the  syml 
divided. 

All  the 
(if  roots,  ha 

Def. 
of  a  root 

Exam 

or,  in  the 

In  or 


m  m 


IltliA  riONAL    bJXPHKSSrONS. 

8.     (I'lb  P. 


159 


7.     a^P. 

m 

9.     {a  +  /y)"  (r?  —  A)-»  10.     /r  »^«. 

II.     «  «^'\  12.     ^-     -•'''-„-, 

(^  — y)  '^ 

KlhIucc  to  simple  products  and  fractions: 

X^ij  "7  .  14.     {(nfj^^c  ")'*. 

15.     (rr^i")''-  16.     Vf    "/    *?. 


17 


18. 


■♦♦»• 


CHAPTER    III. 

REDUCTION    OF    IRRATIONAL    EXPRESSIONS. 


Definitions. 


180.  D(f.  A  Rational  Expression  is  one  in  wliicli 
the  symbols  are  only  added,  subtracted,  multiplied,  or 
divided. 

All  the  operations  wo  have  hitherto  considered,  except  the  extraction 
of  roots,  have  led  to  rational  expressions. 

Def.  An  expression  which  involves  the  extraction 
of  a  root  is  called  Irrational. 

Example.     Irrational  expressions  are 

or,  in  the  language  of  exponents, 

a^,  [a  +  b)K  27i 

In  order  that  expressions  may  be  really  irrational, 


100 


inn.  1  TioNA  L  icxmEssroxs. 


hi, 


\W 


they  must  lu'  Irreducible,  tliat  is,  incapable  of  beiiif: 
t'Xprcsscd  witliout  tlio  radical  sign. 

KXAMl'l-E.      The  CXJU'C'SsioilH 

mv  not    properly  irratioiiiil,  hccausc  they  are  equal  to  a  +  // 
and  0  respectively,  whicli  are  rational. 

7V/'.  A  Surd  is  the  root  which  enters  into  an 
irrational  ex])ression. 

KxAMPLE.  The  expression  a  -f  bVx  la  irrational,  and  the 
surd  is  's/x. 

Def.  Iri-ational  terms  are  Similar  wlien  they  con- 
tain the  same  surds. 

Examples.  The  terms  v^HO,  7v^30,  {x  +  ;/)  a/;}0,  are 
similar,  because  the  ((uantity  under  the  radical  si^i^ni  is  30  iu 
each. 

The  terms  {n  +  V)  ^/x  -\-  y,  3Va:  +  y,  m'^x  +  y  arc 
similar. 

AjfftTej»ati()ii  of  Similar  Teniis. 

181.  Irrational  terms  maybe  aggregated  by  the  rales  of 
^§  5-t-5G,  the  surds  being  treated  as  if  they  were  single  sym- 
bols.    Hence : 

iy7i('n  similar  iin'atiniutl  trnnfi  arc  connected  hy  the 
signs  -}-  or  —,  the  corjficicnts  of  the  similar  surds  mdij 
he  added,  and  the  surd  itself  affixed  to  their  sum. 

Example.    The  sum 

aV{x  -f  y)  -  h\/{x  +  y)  -f  ^^/{x  +  tj) 
may  be  transformed  into  (r?  —  J  -f  3)  \/{x  -f  y). 

exerci.es. 

Reduce  the  following  expressions  to  the  smallest  number 
of  terms : 

I.     7a/3  -  5^/2  +  G\/3  -f  lahV^. 


2.      r, 


3- 

4.     ( 

5-     ^ 

C.     (I 

'•  '! 

8.    '-! 

4 

9-     \ 

lO.       ^ 

II. 


12.       4 


18*2 

so  as  to 
the  metl 

The* 
is  eqiia 

Intl 


Proc 
n^^  pow« 


EXA 


iiiJiA  rioNA  L  Exi'ii  Kssroxs. 


HU 


4 

5 
6 

7 
8 

9 

lO 

II 

12 


-  3  (a  +  b)  V{x  -  d). 

(^/  -f  /v)  's/j'ij  -f-  (rt  —  ^)  y/xy. 

Vx  {a  ~  h)  4-  {fi  —  r)  \/x  +  {r  —  a)  \/x, 
a'\/x  —  ^x  +  "ia^/x  —  {a  -y  b)  y/x, 

'.  y/x  —  ay/x  -\-  (jVx  —  c^/x  -f-  „-  y/x. 
4  J 

—~-  Vx  —  QWx Vx  +  V.-r. 

3    /  /         ,         ,,    /        2  (a  —  A)    / 

^  ^/x  _  Va:  +  (rt  -  />)  V  .'•  +  -^Tj-    -  V  a:- 

V«  —bVa—\/x+  -^ — ^  Vfl  —  .,  va. 

4  ** 

^  V^a;  —  Vx  -\-  -^g — -'  Va;. 
Wx  —  -  \/^-  +  {a  —  b)  Vx. 


Faetoriiiff  Surds. 

183.  Irrational  expressions  may  sometimes  he  transformed 
so  as  to  have  different  expressions  under  the  radical  sign,  hy 
the  method  of  §  178,  applying  the  following  theorem: 

Theorem.  A  root  of  the  product  of  several  factors 
is  equal  to  tlie  product  of  their  roots. 

In  the  language  of  Algehra, 

Vabcd,  el  .  =  Vci  Vb  Vc  Vd,  etc. 

_  ffk  f,k  f,k  flk^    (3tc. 

Proof.  By  raising  the  members  of  this  equation  to  the 
n^^  power,  we  shall  get  the  same  result,  namely, 

a  X  b  X  c  X  d,  etc. 

Example.     -y/SO  =  Vo  Vs. 
11 


103 


Hi  HA  TIONA  L    £XrilESS!0N6. 


EXERCISES. 

Prove  tlic  following  equations  by  computing  both  sides; 

Proof.      V4  a/40  =  2-7  =  14,  and  \/lOG  =  14. 

V4  a/'J  =  a/.'30. 
1/4  V^'o  =  \^4.i>5. 

a/u  a/IG  =:  v'iirTf;. 
A/;i5  a/^jo  =:  v;^rj.;3o. 

ExpresH  with  a  single  surd  the  products* 

I.     V{a  +  b)  V{(i  —  b). 

SoLiTiox.     ^/{a  +  b)  V(a  -  b)  —  \/T(i'^b)  {a  —  b) 

2.    Vt  V'),  3. 

4.    V^  VC^  +  y).  5. 

6.  V(.<:  f  1)a/(.^--1). 

7.  V(.r2 +!).</(./•+  I)  \/(.f- 

8.  |>  +  /.)l(r/_/.)lJ''. 

9.  [(.r2 +!)?.(.•  4.  l)^(.r-l)^.]' 


A/rt  V'^  V{(i  +  />). 
1). 


IS.*?.  If  wo  can  s«^parate  the  quantity  under  the 
radical  sii^n  into  two  factors,  one  of  wliicli  is  a  perfect 
square,  wi»  may  extract  its  root  and  affix  the  surd  root 
of  the  remaining  factor  to  it. 

EXAMPLES. 

V^  =  Va"^  y/b  —  aVb. 
y/ab  \ar  — _  \/ifibc  =  aVbc. 
\/l3  a/h  =  A/r-2  =  A/3n  V3  =  Ga/2. 
a/(4«='  +  S(Cb  -  Ki^r^r)  -.  v/4^^  (^+"  2^— lac) 

—  2^a/(^  +  "lb  —  4ac). 
{x^—  4.r//  f  4,yv/-)  ^  =  (.r  —  2^)  x^. 


IliMA  TIONA  L    EXPRESSIONS. 


1(53 


EXERCISES. 

Reduce,  when  possible  ; 

I.    Vs. 

2. 

a/3:.». 

3.    Vi-is. 

4. 

Va  V2r. 

5.     Vab  Vca  y/hc. 

6. 

V  ^  V72. 

7.    \/4  ^/rz, 

8. 

V'l.r  +  1)  V(.r  +  1). 

9.     VlTo. 

10. 

Viso, 

II.     a/108. 

12. 

V.t'^  («  +  b). 

13.  \/(«'^.^'  +  "lahx  +  <5'\?;). 

Here  the  quantity  under  the  radical  sign  is  equal  to 
(a-  +  2a6  +  W)x  —  {a  A-  b)-  x. 

In  questions  of  this  class,  the  beginner  is  apt  to  divide  an  expression 
like  \/a  +  h  +  c  into  -y/^/  +  y'ft  +  y'r,  which  is  wrong.  Tlie  w|uare 
root  of  the  sum  of  several  quantities  cannot  be  reduced  in  this  way. 

14.  Va^!/ -\- ^ai/ -]- 4:y.  15.     \/-^ifih  ■i-''s7nz~+lz. 
lleducc  and  add  the  following  surds: 


16.     4^2-0^8  +  10^32.     17.     a/12  +  ^27  +  VTo. 
18.      V4a  —  2\/«.  19.     125^  —  45s  —  80I 

20.     a/81  —  V102.  21.     {aW)^  4-  (r72r«)'. 


Multiplication  of  Irrational  Expressions. 

184.  Irrational  polynomials  may  be  multi])lied   by  com- 
bining the  foregoing  principles  with  the  rule  of  §  78. 
The  following  are  the  forms  : 

To  multiply  a  +  h^/x  by  m  -f  nVy. 

a  {m  +  nVi/)  =  uni  x  nn^/ij. 
hy/x{m  +  n^/y)  =  hm^/x  -f  buVxy. 

The  product  is  am  -f  nnVy  +  bmVx  -f  hiVxi/. 

EXERCISES. 

Perform    the   following   multiplications   and    reduce    the 
results  to  the  simplest  form  (compare  §  80) : 

I.     (2  +  3V'5)(5-aV2).         2.     (74-2\/S2)(9-5\/2). 


.  { 


164 


JRItA  TIONAL   EXPRESSIONS. 


3.     {a  +  Vh)  {n  —  vVy).         4.     {Va+  ^/h  +  a/^  +  \/ d)\ 

7.  («  +  a~^)^.  8.     (rt^  —  a  i')^. 

9.  \(i  +  SV(a:  +  2/)]  \ct  —  bV{x  +  2/)]- 

10.  [w  -f  wa/(«  -|-  ^)]  ['«  —  wVi^  —  ^)]. 

11.  {x+  \/{x^  —  1)]  [x  -  ^/{x^  -  1)]. 

12.  l{b'^J^l)^  +  b]  [(i2  +  l)i-^]. 

Expressions  may  often  be  transformed  and  factored   In- 
combining  tbe  forc'going  processes. 

Example.     To   factor  ax^  +  hx^  +  cx^  +  dx^,  we  notice 

tbat  7  1  o  fi  1  n       J 

a:2"  =  a;=i;i-3,        tr^  —  a-^a:^,     etc. 

so  tliat  the  expression  may  be  written, 

ax^x's  -\-  hx^x-  -\-  cxx-  +  dx-  =  {ax^  +  bx^  -f  ex  -f-  d)  x^. 

EXERCISES. 

Reduce  the  following  expressions  to  products: 

13.  2  +  ^^.  14-     32-f2.3i 
15.     {a^r  hf.      ^                  16.     Vi/  4-  ay^'^hf, 

11  —  V^^' 


17.     X  —  }j  —  V  .r  —  y. 
Reduce  to  the  lowest  terms 


18. 


21. 


2 

«  —  3:  4-  Vfl^ 


19. 


A/ff  4-  & 
a  4-  6 


a: 


« 


—  \/« 


20. 


22. 


«^ 


^  4-  bx^~ 


B  ,      1 

aa;^  —  ox^ 


X 


V  a^  —  0 

a  ^  b 


w 


18.1.  Rationalizing  Fractions.  The  quotient  of 
two  surds  may  be  expressed  as  a  fraction  with  a  rational 
numerator  or  a  rational  denominator,  by  multiplyiL, , 
both  terms  by  the  proper  multiplier. 

\/5 


Example.     Consider  the  fraction 


MuH 

Mult 
numerat 

The 
rationa' 
both  of 

Let  I 


in  which 
or  nume: 
merator 
will  beco 

The] 
so  that  i 


Redui 
denomin; 


I. 


V7 


IRRATIONAL    EXPRESSIONS. 


IGi) 


Multiplying  both   terms   l)y   V?,   the    fraction    becomes 

/or 

—^ ,  and  has  the  rational  denominator  7. 


Multiplying  by  y'5,  it  becomes  — — 
numerator  5.  '^ '^'^ 


,  and  has  the  rational 


The  numerator  or  denominator  may  also  be  made 
rational  when  they  both  consist  of  two  terms,  one  or 
both  of  which  are  irrational. 

Let  us  have  a  fraction  of  the  form 

in  which  the  letters  A,  D,  P,  Q,  and  R  stand  for  any  algebraic 
or  numerical  exjiressions  whatever.  If  we  multiply  both  nu- 
merator and  denominator  by    P  —  Q^ R,    the  denominator 

will  become 

P2  -  Q^R. 

The  numerator  will  become 

AP  +  PD\^B  -  A  QVR  -  DQy/~BR. 

so  that  the  value  of  the  fraction  is 

AP  -\r  PD^/B-  A  QVR  -DQVBR 
in_  Q2j[i 


I        ( 


EXERCISES, 


Reduce  the  following  fractions  to  others  having  rational 
denominators: 


4. 

7. 


7\/3 
9\/5" 
a  +  Vb 
a  —  Vb 


2.     — 

5- 
8. 


3\/6  * 
"  —  Vx 

a  -if  -v/a; 


3- 
6. 


bVu 

2\/2 

V'x  —  Vy 


100 


10. 


PEIiFI'JCT  SQUAR/^S. 
a  +  2V(x  +  y)  . .      ''2 V3  -{-  TV's 


n5  +  \/(a;  +  y) 


II. 


Vj  —  V3 


X 


^/x^ 


a* 


14. 


«"  +  (a  +  1)2 


15- 


.2; 


a 


\/'x  +  g  +  V- 

V^  +  ft  —  V2;  —  ft 


Perfect  Squares. 

l^>(>.  Def.  A  Perfect  Square  is  an  expn^ssion  of 
wliicli  the  square  root  can  be  formed  witliout  any  surds, 
except  sucii  as  are  already  found  in  the  expression. 

Examples.  Am\  hfi  +  4ft  +  1  are  perfect  squares,  be- 
cause tlieir  square  roots  are  'Zm\  'Za  +  1,  expressions  without 
the  radical  sign. 

The  expression  a  -\-  "Z^/ah  +  b,  of  which  the  root  is 

may  also  be  regarded  as  a  perfect  square,  because  the  surds 
Vft  and  V^6  are  in  the  ])roduct  2\^ab. 

Cntcrion  of  a  Pfrfcct  Square.     Tiie  question  whether  a 
trinomial  is  a  perfect  s((uare  can  always  ha  decided  by  compa. 
ing  it  with  the  forms  of  i<  80.  namely: 

cfi  -f  2ftZ>  -i-  b^  —  (ft  +  h)\ 


or  ft/2  —  2((b  -\-h^  —  {(    ~  h)\ 

We  see  that  to  be  a  perfect  square,  a  trinomial  muit*^  fulfd 
the  following  conditions: 

(1.)  Two  of  its  three  terms  must  be  perfect  squares. 

(2.)  Th(»  remaining  term  must  be  equal  to  twice  the 
product  of  the  square  roots  of  the  other  two  terms. 

AVlien  these  conditions  are  fuliilled,  the  square  root 
of  the  t''*viomial  will  be  the  sum  or  dilf(»rence  of  the 
square  roui^^  of  tiie  terms,  according  as  the  product  is 
positive  or  negative. 

The  rC'*  may  bav^  eitlier  sign,  'jecause  the  squares  of  posi- 
tive and  ncu".?'*vo  q.iuntitics  have  tlie  same  sign. 


If  tl 
the  trin 


Fine 
and  ext 

I. 

3- 

5- 
7. 
9- 

T  1. 

n- 

IS- 


such a 

the  trii 

Thi 

Pro 
s(piare, 
term,  S( 

Adc 


COMPLETING    THE  SQUARE. 


167 


If  tlie  torms  whieli  are  perfect  squares  arc  both  negative, 
the  trinomial  will  be  the  negative  of  a  perfect  scpiare. 

EXAMPLES. 

^/TF+Jab'^  b^  =  a  +  b  or  —  (a  +  b). 

Va^  —  'Zab  -\-  b'^  =  a  —  b  or  b  —a. 

_  a^  _|_  2ab  -b'^=  -  {a  -  b)^  =  -  (b  -  af. 

EXERCISES. 

Find  which  of  the  following  expressions  are  perfect  squares, 
and  extract  their  square  roots: 


I. 

9  -f  1^  +  4. 

2. 

:c'^  +  Ax  4  •  4. 

3- 

4^1  1  2x^  +  '^.' 

4. 

^/2  _^  ab  —  W. 

5- 

4^^^  +  Ua'^b"  +  0//». 

6. 

(fi  -|_  2ab  —  W. 

7. 

x^  —  aj^ij  +  ,  ah/. 

8. 

a^b^  -  2abcfl  +  (^(P. 

9- 

m  -f  2mhi^  +  w. 

10. 

a'i  —  2nx  +  1/. 

1 1. 

a  +  Aa^b'^  +  4J. 

12. 

a      2  +  (r\ 

13. 

25^/1  +  r»22  _  ^OjS^q. 

14. 

(\hni^n  ^  7^2  ^  9„,4/i, 

TC. 

40r2;/2  4_  0>2         ^-rruv.- 

t6. 

Oj»8«  —  'Jm^'hni  -I- '    ^ 

To  Complete  the  Square. 

187.  If  one  term  of  a  binomial  is  a  perfect  square, 
such  a  tenn  can  always  be  added  to  the  binomial  that 
the  trinomial  thus  formed  shall  be  a  perfect  square. 

This  operation  is  called  Completing  the  Square. 

Proof.  Call  a  tlie  root  of  the  term  which  is  a  perfect 
S({uare,  which  term  wo  suppose  the  firs f,  and  call  >n  the  othc" 
term,  so  that  the  given  binomial  shall  be 

a^  +  m. 

Add  to  this  binomial  the  term    ,   . ,  and  it  will  become 

4fr 


a^  -\-  m  -f 


4a^' 


1G8 


COMPLETING    TUE  St^F  [RE. 


•^    * 


This  is  a  perfect  square,  namely,  the  square  of 


«■  +  : 


that  is, 


a'  +  m  + 


4a2 


Hence  tlie  followiiief 


Rule.     .Idd  to  the  hinoniial  the  square  of  the  secoiul 
term  divided  bjj  four  times  the  first  term. 

Example.     What  term  must  be  added  to  the  expression 

3?  —  ^ax  (1) 

to  make  it  a  perfect  square  ? 

The  rule  gives  for  the  term  to  })e  added, 

{—4:(txf 


4txi 


4:a\ 


Therefore  the  rc(|uired  perfect  square  is 

a;2  _  ^ax  +  4r^2  =  {x  —  2a)\ 

We  may  now  transpose  4«2^  so  that  the  left-hand  member 
of  the  equation  shall  be  the  original  binomial  (1).     Thus, 

a;2  —  ^ax  —  {x  —  2aY  —  4^2. 

The  original  binomial  is  now  expressed  as  the  difference  of 
two  S(iuares.  Therefore,  the  above  process  is  a  solution  of  the 
problem :  Having  a  bi?iomial  of  irhich  07ie  term  is  a  perfect 
square,  to  express  it  as  a  difference  of  tivo  squares. 

EXERCISES. 

Express  the  following  binomials  as  differences  of  two 
squares : 


I.  x^  +  2xy. 

3.  x^  -\-  Qax. 

5.  A:X^  -\-  4'ry. 

7.  Ux^  +  ^2mx. 

9.  «2«2  _|_  2a2a;. 

II.  m^x'^  4-  1. 


7.  X?  -\-  4:xy. 
4.  4:^2  -|-  4:xy, 
6.  dx^  +  ax. 

8.  x^  -f  4x. 
10.  Ir^x^  +  2. 
12.  9p^j(^  4-  bx. 

Ga\ 


14 


1 


JliliA  TIONA  L    FA  CTORS. 


169 


IrraticMiiil  Factors. 

•  188.  When  we  introduce  surds,  many  expressions  can  bo 
factored  which  have  no  rational  factors.  The  following 
theorem  may  be  ap])lied  for  this  })urpose : 

Theorem.  Tlie  difference  of  any  two  quantities  is 
eqnal  to  the  product  of  sum  and  difference  of  tlieii' 
square  roots. 

In  the  language  of  algebra,  if  a  and  h  be  the  quantities,  we 
shall  have 

which  can  be  proved  by  multiplying  and  by  §  80,  (3). 


Factor 


EXEF^CISES, 


I.     m  —  71. 
3.     a?n  —  hn. 


2.  m  —  1. 

4.  ^ahn  —  0. 

5.     x^  —  m.  6.  x^  —  {m  -(-  «). 

7.     {x  —  «)2  —  J  (m  —  7i).      8.  x^  —  {m  —  w). 

Find  the  irrational  square  roots  of  the  following  expressions 
by  the  principles  of  §  186  : 

11.  a  —  2  -{-  a~K  Ans.  a^  —  a~K 

12.  X  —  2x^-1'!/  +  l/'              13.  4  -f-  -iVS  +  3. 
14.     9  4- 5  —  r)\/5.                 15.  'Ui  -\-  f)  — -^ahK 
16.     a-^b-\-2{a-^b)h--{-x\    17.  3  -\-  -.'^15  +  5. 


18.     3-1-  T)  —  2  a/15. 

20.    a  —  %y/a  -f  1. 

i       1 

22.     a  -\  2rt*  -I-    i- 
as 

M^     jfg  +  g- 
a6.     a"  -H  2  -f-  a-^. 

t8.     rr  +  />  —  4  + 


ai.     «  —  2a*^  +  as. 
23. 


7 


4 


4_ 


''•     16 +  4"^  4* 
27.     4jr3  —  8  +  4a:-«. 


t 


r 


BOOK    V  I. 

EQUATIONS   REQUIRING  IRRA 
TIONAL     OPERA  TIONS. 


I  4 


CHAPTER    I. 
EQUATIONS    WITH    TWO    TERMS    ONLY. 

189.  la  the  present  eluipter  we  consider  equations  which 
contain  only  a  sinijle  i)ower  or  root  of  tlie  iinknowii  (juantity. 

Such  an  e(iuation,  when  reduced  to  tlie  nornuil  form,  will 
be  of  tlie  form 


J.r«  +  B 


0. 


By  transposing  />*,  dividing  })y  A,  and  puttin 


or 


B 

the  equation  may  be  w  xitten, 

ocP'  —  a  =  0. 

or  X"'  —  a,  (1) 

Here  n  may  be  an  integer,  or  it  may  represent  some  fraction. 
Such  an  equation  is  called  a  Binomial  Equation,  because 
the  expression  ocP'  —  a  is  a  binomial. 

Solution  of  ii  Biiioniial  Equation. 

190.  1.  When  the  exjwnenf  of  x  is  a  {Dhole  number.  If  we 
extract  the  ?i^^  root  of  both  members  of  the  equation  (1),  these 
roots  will,  by  Axiom  V,  still  be  equal.  The  n^^  root  of  :r«  being 
Xj  and  that  of  a  being  a",  we  have 


and  the  equation  is  solved. 


X 


or 


,^*5I.  •  • 


BIND  MI  A  L    Eq  UA  riUN8. 


171 


8.   When  the  cxjwucnt  is/ractionuL    Lut  Ihu  efiuatioii  be 

m 
a^  =  a. 

Raising  both  menibers  to  tho  n"^  power,  wc  havo 
Extracting  tlio  7«"^  root, 

n 
X  =  </'". 

If  the  numerator  of  tlio  exponent  is  unity,  we  only  have  lo 
suppose  ?»  =  1,  wliieli  will  give 

X  ■=  a"'. 

Hence  the  ])inoniial  ecpnitioii  always  admits  of  solution  by 
forming  powers,  extracting  roots,  or  both. 


Special  Foriiis  of  Biiioiniul  Equations. 

Def.  AVhen  tlie  exponent  7i  is  an  integer,  the  e(iiia- 
tion  is  called  a  Pure  Equation  of  the  decree  n. 

When  71  =  2,  the  equation  is  a  Pure  Quadratic 
Equation. 

When  /i  =  3,  the  equation  is  a  Pure  Cubic  Equa- 
tion. 

EXERCISES. 


Find  the  values  oi  x  in  the  following  e([uations 


P 

x^ 


=  n- 


Ans.    X  = 


P' 


a  -\-  b 


a 


=  c. 


9 


X 


X* 


7? 


8. 


X 


X^ 


yl 


•2  


mi 
a 


nx^  —  () 


.r^  - 

-b 

X  — 

2a 

X  — 

■  a 

<i  + 

b 

m 

— " 

:cn 

j:"  —  a 
2x—  b 

. —  ■  -—  ■  « 

x  —  b 
p 


a 


r 


x^ 


a' 


a  +  b 


b  —  a 


V^ 


a* 


172 


POSITIVE   AND   NKdATIVK   ROOTS. 


In  tho  Inst  oxnnii)l«,  cleariog  the  equation  of  fractionn,  we  shall  have 

V'j'  -  rt*  =  6«  -  «», 
or  (T'  -  n*)^  rn  i«  -  a\ 

We  Hfiiiiirc  both  sides  of  this  fquution,  which  gives  another  In  which 
'jt-  only  apix-ars. 

lo.     (./•  —  ^/):'  =  b^.  II.     (/^  —  (fi)^  =  mx. 

12.      {'^X  —  V/5»)^  =  UXK 

Positive  and  Xoffativc  Roots. 

101.  Since  tlic  s(|U{ire  root  of  a  quantity  may  bo  eitlier 

positive  or  negative,  it  follows  that  when  we  have  an  equation 

such  as 

^  =  a, 

and  extract  the  square  root,  we  may  have  either 

X   =     +  a5, 

or  X  z=  —  aK 

Hence  there  arc  two  roots  to  every  such  equation,  the  one 
positive  and  the  other  negative.  We  express  this  pair  of  roots 
by  writing 

X  =  ±  nK 

the  expression  ±  ««  meaning  eitlier  +  n^  or  —  ah 

It  might  seem  that  since  the  S(iuare  root  of  x"^  is  either  +.r  or  —a*,  we 
should  write 


having  the  four  equations, 


±  .T  =  ±  a*, 
X  =  a*, 
X  ■=  —  «*, 

—  X  =    +  rt*, 

—  X  =  —  tiK 


But  the  first  and  fourth  of  these  equations  give  identirnl  values  o.  x 
by  simply  changing  the  sign,  and  so  do  the  second  and  third. 

PROBLEMS    LEADING    TO    PURE     EQUATIONS. 

1.  Find  thro'  numbers,  such  that  the  second  shall  be 
double  the  first,  the  third  one-third  the  second,  and  the  sum 
of  their  squares  19G. 

2.  The  sum  of  the  squares  of  two  numbers  is  3G9,  and  tho 
difference  of  their  squares  81.     What  are  the  numbers? 


J'Ji(>llLh\)fS. 


ITA 


3.  A  lot  of  hud  rniituins  Uit,")  f!f|u;iro  foci,  imd  its  l('n.i;(!» 
rxrocMJs  its  breiulth  hv  1'^  tVct.  What  arc  tljo  length  iiud 
bivadll:? 

To  solve  tills  equation  ns  u  biiionrial.  take  the  mean  of  the  lengtli 
and  breadth  as  tht;  unknown  <iuantity,  ho  that  the  lenj^th  Bhall  be  ar*  inueh 
greater  than  tin."  mean  uh  the  breadth  is  less. 

4.  V\ui\  a  iiuii»l)cr  siicli  that  if  !»  l)c  added  lo  and  sulitractcd 
from  it,  thi'  product  of  the  siiiii  and  difl'i  iviicu  .shall  ho  ITo. 

5.  Find  a  inuiibor  such  that  if  rHtc  added  to  it  and  suh- 
tractod  from  it  the  product  of  tlio  sum  and  dill'croncc  shall  ho 
'^((  +  1. 

6.  One  numhor  is  double  another,  and  the  diltcn?ncc  of 
their  s(|uaro8  i.s  lU'^.     Winit  are  the  numbers  ? 

7.  One  number  is  8  times  another,  and  the  sum  of  their 
t'ube  roots  is  1'^.     What  are  the  nund)er.s  ? 

8.  Find  two  numbers  of  which  one  is  3  times  the  other, 
jind  the  s(|uare  root  of  their  sum,  multiplied  by  the  lesser,  i.s 
e(|ual  to  I'iS. 

9.  Wiuit  two  numbers  are  to  each  other  as  2  :  3,  and  the 
sum  of  thi'ir  scpuires  =  [i'i^y? 

Note.     If  we  represent  one  of  the  numbers  by  2/,  the  other  will  be  3x. 

10.  What  two  numbers  are  to  each  other  as  in  :  n,  and 
the  s(|uare  of  their  dilferenee  equal  to  their  sum  ? 

11.  What  two  numl)ers  are  to  each  other  as  1)  to  7,  and  the 
cnbe  root  of  their  ditferenee  niulti{)lied  by  the  square  root  of 
their  sum  e(|ual  to  1(!  ? 

12.  Find  j:  and  y  from  the  expiations 

ax^  -\-  liif^  =  c, 
a'x^  +  b')/  =  c'. 

13.  The  hy])otlienusc  of  a  ri<;ht-aup:lcd  trianirlo  is  20  feet 
in  length,  and  the  sum  of  the  sides  is  34  feet.     Find  each  si<U'. 

Note.  It  Ih  shown  in  Geometry  that  the  wjuafe  of  the  hyi)othenuso 
of  a  rifjht-angled  trian<rle  is  equal  to  the  sum  of  the  squares  of  the  other 
two  sides.  In  the  j)resent  problem,  take  for  the  unknown  i|uantity  tin; 
amount  by  which  each  unknown  side  ditters  from  half  their  .sum. 

1.4.  Two  ]K)ints  start  out  tofrethcr  from  the  vertex  of  a 
right  angle  along  its  res})oetivc  sides,  the  one  moving?//  feet 
])er  secoiul  and  tlie  other  71  feet  per  second.  How  long  will 
they  recpiire  to  be  c  feet  apart? 

15.  By  the  law  of  falling  bodies,  the  distance  fallen  is  pro- 
portional to  the  sfpiare  of  the  time,  and  a  body  falls  IG  feet 
the  lirst  second.     How  long  will  it  require  to  fall  h  feet? 


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174 


QUADRATIC  EQUATIONS. 


I' 


CHAPTER     II. 
QUADRATIC     EQUATION  S. 

103.  Def.  A  Quadratic  Equation  is  one  wliicli, 
wlien  reduced  to  the  normal  form,  contains  the  second 
and  no  higher  power  of  the  unknown  quantity. 

A  quadratic  ecjuation  is  the  same  as  an  equation  of  the  second  degree. 

Def.  A  Pure  quadratic  equation  is  one  which  con- 
tains the  second  power  only  of  the  unknown  quantity. 

The  treatment  of  a  pure  (quadratic  equation  is  given  in  the  preceding 
chapter. 

Def.  A  Complete  quadratic  equation  is  one  which 
contains  both  the  first  and  second  powers  of  the  un- 
known quantity. 


The  normal  form  of  a  complete  quadratic  equation  is 


rt.?;2  -f-  ^^  +  c  = 

0. 

If  we  divide  this 

equation  by  a,  we 

obtain 

h         c 

a?  ^  -x-\-  -  = 
a         a 

0. 

Putting, 

for  brev 

h 
c 

<1) 


(2) 


9 


thvi  equation  will  be  written  in  the  form, 

x^  +  2^^  -\-  q  —  0.  (3) 

Def.    The  equation 

(V^  ■{-  px  +  q  —  0 

is  called  the  General  Equation  of  the  Second  Degree, 
or  the  General  Quadratic  Equation,  because  it  is  the 
form  to  which  all  such  equations  can  be  reduced. 


QUADRATIC   EQUATIONS. 


175 


Solution  of  a  Complete  Quadratic  Equation, 

193.  J.  quadratic  equation  is  solrcd  by  adding  such 
(I  quantity  to  its  two  niemhcrs  that  the  niemher  contain- 
iii<;>   the   unknown  quantity  sliall  he  a  perfect  square. 

(§m.) 

\Vc  first  transpose  q  in  the  general  equation,  obtaining 
x^  -^  px  -=1  —  q. 

We  then  add  ^  to  both  members,  making 

x^  ^px  -^-'-^  ='-  —  q. 

The  first  member  of  the  equation  is  now  a  perfect  square. 
Extracting  the  square  roots  of  both  sides,  we  have 


...f  =  ± 


7i 


From  this  equation  wc  obtain  a  value  of  x  which  may  be 
put  in  either  of  the  several  forms. 


.  =  -|± 


n~ 


-1- 


X 


X 


p       Vp^  —  '^q 


2 


2 


=  '^{-p±  Vp^-^q)' 


If  instead  of  siibstituting^  and  q,  we  treat  the  equation  in 
the  form  (3)  precisely  as  we  liave  treated  it  in  the  form  (8),  we 
shall  obtain  the  several  results, 


1  h^       1  J2 


X^  ^  -X  -{-  -r-,  = 


a 


4  a^       4  ci^ 


«■ 


and 


*  =  -2^,± 


/(5  -  «) 


_  —b±  V(h-  -  iac) 


J-;:  1:1 


»^r.., 


I 


176 


q  UADltA  TIG  Eq  UA  TIONS. 


4 


'J  4 


i       '* 


194.  The  equation  in  the  normal  form,  (1),  may  also  bo 
solved  by  the  following  process,  wliich  is  sometimes  more  con- 
venient. Transposing  c,  and  multiplying  the  equation  by  a, 
we  obtain  tiie  result 

a'^x^  +  ahx  = 


ac. 


Z.2 


To  make  the  first  member  a  perfect  square,  we  add  -  to 

each  member,  giving 

U^       ^ 

a^x^  4-  ahx  -\-  --  —  ~ ac. 

4        4 

Extracting  the  square  root  of  both  sides,  we  have 

from  which  we  obtain  the  same  value  of  x  as  before. 

195.  Since  the  square  root  in  the  expression  for  x  may  be 
either  positive  or  negative,  there  will  be  two  roots  to  every 
quadratic  equation,  the  one  formed  from  the  positive  and  the 
other  from  the  negative  surds.  If  we  distinguish  these  roots 
with  .Tj  and  a^g,  their  values  will  be 

—  -  ^  +  Vjb'^  -  4:nc) 

';  }  W 


x^  = 


x^  — 


2a 


We  can  always  find  the  roots  of  a  given  quadratic  equation  by  sub- 
stituting the  coefficients  in  the  preceding  expression  for  x.  But  the  stu- 
dent is  advised  to  solve  each  separate  equation  by  the  process  just  given, 
which  is  embodied  in  the  following  rule  : 

I.  Reduce  the  equation  to  its  normal  or  its  general 
form,  as  may  he  most  convenient. 

II.  Transpose  the  terms  which  do  not  contain  x  to  the 
second  member. 

III.  //  the  coefj/iciejbt  of  x^  is  unity,  add  one-fourth 
the  square  of  the  coejficient  of  x  to  both  members  of  the 
equation  and  extract  the  square  root. 

IV.  //  the  coejficient  of  x''^  is  not  unity,  either  divide 
by  it  so  as  to  reduce  it  to  unity^  or  multiply  all  the  terms 


w 


to 


(4) 


QUADRATIC   EQUATIONS. 


Ill 


hy  such  a  factor  that  it  shall  become  a  perfect  square, 
and  complete  tlie  square  hij  the  rule  of  §  187. 


Solve  tlie  equation 


X 


EXAMPLE, 
—  1 


=  'Zx. 


Clearing  of  fractions  and  transposing,  we  find  the  equation  to  become 
2a;2  -  412;  +  1  =  0,  (5) 

„       4Ax   _  _1 
^       '"'Z    ~~        2' 
Adding  |  the  square  of  the  coefficient  of  x  to  each  side,  we  have 


2       ^^        , 


1G81        1081 


1 
2 


1G73 


2  "^    '      10    ~     10         2  10 

Extracting  the  square  root  and  reducing,  we  find  the  values  of  x  to  be 

:,  :=  ^  (41  +  ^1073), 


and 


X. 


X. 


=  l(41-\/lG73). 


Using  the  other  method,  in  order  to  avoid  fractions,  we  multiply  the 
equation  (5)  by  2,  making  the  equation, 

4:X^  —  82:c  =  —  2. 

Adding  —  =  ——■  to  each  side  of  the  equation,  we  have 

412        1681       ^        1673 

4:^2  _  82a;  +  -^  =  -4-  -  ^  =  -^' 

Extracting  the  square  root. 


v.'honce  we  find 


^^—     2      —    V         4:         ~~  2  ' 

41  ±  a/1673 


X  = 


the  same  result  as  before. 


EXERCISES. 

Reduce  and  solve  the  following  equations 

2    x-{-'Z~  q'  '   y-^    y  +  ^     ^ 

12 


I. 


X 


'(  \\ 


i     I 


I"     '  11 


!  I 

il 


e    ' 


178 


Q  UA 1)  IIA  TKJ  EQ  UA 1  IONS. 


«,  ^\ 


3- 

12           4 

a:      1    '  a;      2       3 

4- 

2/2  —  'Zay  -f  rt^  —  ^2  —  0. 

5- 

1                111 

(I  +  1)  -\-  X  ~  a       b       X 

6. 

x^  —  cc      X  -\-  a      X  —  a 

7. 

x  —  a        ^ 

X  —  a 

X  -\-  a 

8. 

2  +  i/       2/2  _  4    •   2  -  y 

9- 

y  -\-  a      y  —  a           1               1               1 
y  —  a      y-^a~y  —  a~~if  —  d^      y  —  a 

'"'              ^       1   3        0 

O. 

a  -\-  X       a  —  X 

PROBLEMS. 

1.  Find  two  numbers  such  that  their  difference  shall  be 
G  and  their  product  507. 

2.  The  difference  of  two  numbers  is  0,  and  the  difference 
of  their  cubes  is  930.     What  are  the  numbers  ? 

3.  Divide  the  number  34  into  two  such  parts  that  the 
sum  of  their  squares  shall  be  double  their  product  ? 

4.  The  sum  of  two  numbers  is  GO,  and  the  sum  of  their 
squares  1872.     What  are  the  numbers  ? 

5.  Find  three  numbers  such  that  the  second  shall  be  5 
greater  than  the  first,  the  third  double  the  second,  and  the 
sum  of  their  squares  1225. 

6.  Find  four  numbers  such  that  each  shall  be  4  greater 
than  the  one  next  smaller,  and  the  product  of  the  two  lesser 
ones  added  to  the  product  of  the  two  greater  shall  be  312. 

7.  A  shoe  dealer  bought  a  box  of  l)oots  for  $210.  If  there 
had  been  5  pair  of  boots  less  in  the  box,  they  would  have  cost 
him  $1  per  pair  more,  if  he  had  still  paid  $210  for  the  whole. 
How  many  pair  of  boots  were  in  the  box  ? 

Rem.    If  we  call  x  the  number  of  pairs,  the  price  paid  for  each  pair 

210 
must  liavc  been  — -  • 


8.  A 

ij^lO,  and 
chickens 
ksri.     11 

9.  A 

lie  had  ,s 
would  h; 
sell? 

10. 

bill  to  1 
ing,  eacl 
up  the  h 

II. 

])roved 
el'  2  cei 
he  buy  ':* 

12. 

if  he  ha( 
him  4  li 
he  travo 

13- 

its   area 

breadth 

14.  ' 
a  feet,  a 

sions  ? 

15- 
hour. 

speed  as 

has  dri> 

is  the  d 

Not] 
as  unknc 


Equal 

19( 

power 
power 
solved 


QUADRATIC   EQUATIONS. 


171) 


8.  A  liiickstcr  Imuf^lit  a  ccrt;iin  nunibor  of  chickens  f<»r 
li^lO,  iiiul  II  nimilxT  of  turkeys  for  >!l5.Tr).  'I'lu-re  were  4  riioro 
rhickeiiH  tliaii  turkeys,  l)ut  tlu\v  .  acli  cost  him  o.'>  cents  a  piece 
li'ss.     How  many  of  eacii  did  he  buy? 

9.  A  farmer  sold  a  certain  numher  of  shei^p  for  $'Mn.  ]f 
lie  liad  sold  a  numher  of  sheep  0  greater  for  the  same  sum,  iiu 
would  have  received  $4  a  i)iece  less.     How  many  sheep  lid  he 

R'll  ? 

10.  A  party  having  dined  together  at  a  hotel,  found  the 
hill  to  be  80.60.  Two  of  the  numher  having  left  before  i)ay- 
ing,  each  of  the  remainder  had  to  pay  :i4  cents  more  to  make 
up  the  loss.     What  was  the  number  of  the  I)ai1y  ? 

11.  A  pedler  bought  $10  worth  of  ap])les.  ,'K)  of  them 
])roved  to  be  rotten,  but  he  sold  the  renuiinder  at  an  advance 
of  2  cents  each,  and  made  a  profit  of  -ii'S.^U.  How  many  ditl 
he  buy  ? 

12.  In  a  certain  number  of  hours  a  man  traveled  48  miles  ; 
if  he  had  traveled  one  mile  more  pt'rhour,  it  would  have  taken 
him  4  hours  less  to  perform  his  journey  ;  how  many  miles  did 
he  travel  pur  hour? 

13.  The  perimeter  of  a  rectangular  field  is  160  metres,  and 
its  area  is  1575  sq_uare  metres.  What  are  its  length  and 
Ijreadth  ? 

14.  The  length  of  a  lot  of  land  exceeds  its  breadth  by 
a  feet,  and  it  contains  m"^  square  feet.  AVhat  are  its  dimen- 
sions ? 

15.  A  stage  leaves  town  A  for  town  B,  driving  8  miles  an 
hour.  Three  hours  afterward  a  stage  leaves  B  for  A  at  such  a 
speed  as  to  reach  A  in  18  hours.  They  meet  when  the  second 
has  driven  as  many  hours  as  it  drives  miles  per  hour.  What 
is  the  distance  between  A  and  B  ? 

Note.  The  solution  is  very  simple  when  the  propei-  quantity  is  taken 
as  unknown. 


Equations  which   may  be   Reduced   to  Quad- 
ratics. 

196.  Whenever  an  equation  contains  only  two 
powers  of  the  unknown  quantity,  and  the  index  of  one 
power  is  double  that  of  the  other,  the  equation  can  be 
solved  as  a  quadratic. 


!•• 


I 


If 


180 


QUAmUTW  EQUATION  a. 


I  I 


'li, 


Special  Example.    Let  us  take  tlic  oquution 


(1) 


Transposing  c  and  adding  .U^  to  each  side  of  the  equation, 
it  becomes 

4  4 

The  first  member  of  this  equation  is  a  perfect  square, 
namely,  the  square  of  a:^  +  -  ^.  Extracting  the  square  roots 
of  both  members,  we  have 


a;3 


-v\i  =  \/'{^^b^   -c)^  ±\  V{b^  -  4c). 


'  =  l[-b±V{l^'-^c)]. 


Hence,  x 

Extracting  the  cube  root,  we  have 

X 


General  Form.  We  now  generalize  this  solution  in  tlie 
following  way.  Suppose  we  can  reduce  an  equation  to  tlie 
form 

ax^^  -f-  bx"^  -]-  c  =  0, 

in  which  the  exponent  n  may  be  any  quantity  whatever,  entire 

or  fractional.     By  dividing  by   a,  transposing,   and  adding 

1  b^ 

to  both  sides  of  the  equation,  we  find 


4a2 


a  4  a^       4  «^      « 


The  first  side  of  this  equation  is  the  square  of 

lb 

Hence,  by  extracting  the  square  root,  and  reducing  as  in 
the  general  equation,  we  find 


x^' 


=  ^j^[-b±  V{b'  -  4a.)]. 


.s 


II  UADliA  TIC   Eq  UA  TIONS. 


181 


Extracting  tlie  m''*  root  of  both  sides,  we  have 


4*^  (r 


={ 


—  b±  y/¥  —^ic\^ 


a 


If  tlic  exponent  w  is  a  fraction,  the  same  course  may  be 
Inllowed. 

►Suppose,  for  example, 

Dividing  by  a  and  transposing,  we  have 


4  5     3  c 

x-^  +     x^  = • 

a  a 


Adding  - j>.  ^^  both  sides. 


4   ,    J    a   ,     J^  U^        c 

xt  J —  x^  -\ = • 


The  left-hand  member  of  this  equation  is  the  square  of 

b 

Extracting  the  square  root  of  both  members, 
!      A  —  (  ^  _^\^  -  (<^^  —  4^g)^ 


whence, 


x^ 


2  _  —b±  (Z>3  —  4:ac)^ 


2a 


Kaising  both  sides  of  this  equation  to  the  |  power,  we  have 

-  b  ±  (b^  -  4.ar)V 


X 


2a 


EXERCISES. 


1.  Find  a  number  which,  added  to  twice  its  square  root, 
will  make  99. 

2.  What  number  will  leave  a  remainder  of  99  when  twice 
its  square  root  is  subtracted  from  it. 


^I» 


rffiA 


'.  fi 


•f 


:•'. 


182 


Q  UA  DRA  TIG   EQ  UA  TfONS. 


^  I 


■f 


^%  Ono-fiftli  of  a  cortiiin  number  exceeds  its  square  root 
>)V  J}<).     Wliat,  is  tlie  nuinbcr? 

4.  What  nunil)ei'  added  to  its  square  root  makes  liOd  ? 

5.  II'  Trom  '.i  times  a  eertuin  number  we  subtract  10  times 
its  scjuare  root  and  9(5  more,  and  divide  the  rcuuiimlcr  by  the 
number,  the  quotient  will  be  2,     What  is  the  number? 

Solve  the  equations : 


6.     li/-2i/^ 


15. 


7.     dy*  -  7^2  =  25. 


8.     5^^  —  Sy^  =  13. 


m 


m 


9.     {x^  +  n^Y  _  4  (a;2  4-  fl52)2«  =  ^^2  _  2  + 


1^ 

a2 


11)7.  AVHien   the   unknown  quantity  ai)pears  in  the  form 

x^  +2,  the  square  may  be  completed  by  simply  adding  2  tu 

^  .  1     . 

this   expression,   because    x^  -\-  2  -\ — ^    is   a    perfect    square, 

1  ^ 

namely,  the  square  ot  x  -\-   -    Tlie  value  of  x  may  then  be 

deduced  from  it  by  solving  another  quadratic  equation. 


Example. 


3x^  +  -,  =  23. 

x^ 


We  first  divide  by  3  and  add  2  to  each  side  of  the  equation, 


obtaining 


0^1  ^2  ^  /CO 

^  +  2  +  :;^  =  T  +  ^  =  T 


Extracting  the  square  root  of  both  sides, 


1        2\/7 
X  -i--  =  —— 


2V21_2    . 
-3~  -  3  ^^^- 


By  multiplying  by  x,  this  equation  becomes  a  quadratic, 
and  can  be  solved  in  the  usual  way. 

Let  us  now  take  this  equation  in  the  more  general  form, 


^  +  -  =  ., 


(«) 


which  reduces  to  the  foregoing  by  putting  6  =  0  V21.    Clear 
ing  of  fractions  and  transposing, 

a^  —  ex  -\-  1  =  0  ; 


whic'l 


Q  UA  DRA  TIC   EQ II A  TIONS. 


183 


wliicli  being  solved  in  the  usual  way,  gives 

e  ±  ^(^2  -  4) 


X  = 


'Z 


The  two  roots  are  therefore 


e  +  ^{i'i  -  4) 


( 


x«  = 


e  _  ^/{c^  _  4) 


2 


If  in  tiic  first  of  these  equations  we  rationalize  the  numer- 
ator by  multiplying  it  by  e  —  V{e^  —  4)  (§  IS-j),  we  siuiU  tiiid 


it  to  reduce  to 


2 


1 


that  is,  to 


X. 


Therefore, 


x^  =:  —  idc7itic((Ufj. 


X, 


x^ 

Vice  versa,  x^  is  identically  the  same  as 

This  must  be  the  case  wlienever  we  solve  an  equation  of  the 

form  ia),  that  is,  one  in  whicli  the  value  of  x  +  -  is  given. 

50 

Let  us  suppose  first  that  e  =  y,  so  that  the  equation  is 

1       50 

X        7 

It  is  evident  that  x  =  7  is  a  root  of  this  equation,  because 
when  we  put  7  for  x,  the  left-hand  member  becomes  7  +  ^ , 

which  is  equal  to  -— •    If  we  put  -^  for  x,  the  left-hand  mem- 
ber will  become 


1        1^ 

7  "^    1 


+  7. 


\  h 


.1  - 


;  \. 


'   I 


»    I 


Hence  x  and  -  exchange  values  by  putting  ^  instead  of  7, 
so  that  their  sum  x -\r  ~  remains  unaltered  by  the  change. 

X 


■i  , 


'!) 


u 


If 


u- 


184  Q  UA  I) HA  TfC   h'Q  VA  7'IOXS. 

The  goiioral  rosiilt  may  ho  expressed  tlius: 


I 


Becuiisi;   the  value  of  the  expression  x  -f-      remains  im. 

1  ' 

altered  when  we  elianpfe  x  into     ,  therefore  the  reciproeal  of 

'C 

any  root  of  the  e([uation 

1 

X  4-  -  =  e 

X 

is  also  a  root  of  the  same  ecpuition. 

EXERCISES. 

Find  all  the  roots  of  the  following'  e<|iiations  wiihoiit  elcar- 
in;::  <he  fj;ivcn  equations  from  denoniinalons: 


3-     ^'^f  +  \f 


17 


2.     <i'^x^  4-  -- -  =  vi^  —  2. 


f(h-^ 


28. 


in' 


4-     -^2-^1/'  =  ^''^'' 


5.  Show,  without  solving,  that  if  r  he  any  root  of  the 
equation  i 

X'' 

then  —  7\     ,  and  —      will  also  be  roots. 
r  r 

Factoring  a  Qiuidraiic  Equation. 
108.    1.  Special  Case.     Let  us  consider  the  equation 

x^  —  2a:  —  15  =  0, 


or 
or 


X* 


2a:  +  1  —  IG  rzz  0, 


(.c  _  1)2  _  42  ^  0. 

Factoring,  it  becomes  (§  90), 

{x-1-^  4)  {x  -  1  _  4)  =  0, 
or  {x  +  3)  (x  —  5)  =  0. 

Therefore  the  original  equation  can  be  transformed  into 

(x  +  3)  {X  -  5)  rr  0, 

a  result  which  can  be  i^roved  by  simply  performing  the  multi- 
plications. 


is    11  !|. 

•cal  (»f 


clcar- 


f^UADHA  TIC    h'Q L/A  '/'IONS. 


IHf) 


This  last  c(iuation  inuy  bo  satirifu'd  by  putting  either  uf  itd 
factors  eqiiul  to  zero  ;  that  is,  by  supponing 

a;  -f  3  =  0,     whciico    x  —.  --  3  ; 

or  j;  —  5  =  0,     whence     a*  —  +  5. 

Tlicsc  are  the  same  roots  which  we  sliouhl  obtain  by  solving 
the  original  equation. 

5.\  FacforitHj  the  General  Qufulratir  Kqnatlon.     Let  us  con- 
siiler  the  general  (quadratic  equation, 

a^  -\-  por  -^  ([  =  0.  {a) 

Now,  instead  of  thinking  of  ./•  as  a  root  of  this  c(|uation, 

lot  us  suppose  X  to  have  any  value  wiiatever,  and  let  us  con- 

siiler  the  expression 

x'  +  px^-q,  (I) 

wiiich  for  shortness  wo  shall  call  T.     Let  us  also  in(|uirc  how 
it  can  be  transformed  without  changing  its  value. 

First  we  add  and  subtract  -. />",  so  as  to  make  part  of  it  a 
perfect  square.     It  thus  becomes, 

1 

V       4 


X=x'^+px^\p'^-\p'i  +  q', 


or,  which  is  the  same  thing. 

Factoring  this  expression  as  in  §  188,  it  becomes 


X  = 


^4-  o 


\p  +  Q;>^  -  rjf^  [x  +  \p-  Q;>2  -  ry)-^]. 


The  student  should  now  prove  that  this  expression  is  really  equal  to 
if  +  px  +  q,  by  performing  the  multiplication. 


Let  us  next  put,  for  brevity, 

1  /I    ,        \\ 


(2) 


» 


>' 


186 


qUADRATIC   EQUATIONS. 


■m 


The  preceding  value  of  X  will  then  hecome, 

X  =  {x  —  «)  (x  —  (3),  (3) 

an  expression  identically  eqnal  to  (1),  when  we  put  for  «  and 
(3  their  values  in  (2). 

Lot  us  return  to  the  supposition  that  this  expression  is  to 
be  etpial  to  zero,  and  that  x  is  a  root  of  the  equation. 

The  equation  (a)  will  then  be 

{x  ~  cc)  {x-13)  =  0.  (4) 

But  no  product  can  be  equal  to  zero  unless  one  of  the  fac- 
tors is  zero.     Hence  we  must  have  either 

X  —  a  =:  0,     whence    x  =  «  ; 
or  X  — 13  =  0,    whence    x  =  (X 

Hence,  u  and  ^  are  the  two  roots  of  the  equation  (a). 
The   above   is    another    way  of    solving    the    quadratic 
equation. 

To  compare  the  expressions  (1)  and  (3),  let  us  perform  the 
multi[)lication  in  the  latter.     It  will  become, 

jr  =  x^  —{((-{-  [3)  X  +  ccf3. 

Sir. '30  this  expression  is  identically  the  same  as  x^-j-px  +  q, 

the   coefficients   of  the  like  powers  of  x  must  be  the  same. 

That  is, 

a-:-f3=~p,) 

which  can  be  readily  proved  by  adding  and  multiplying  the 
equations  (3). 

This  result  may  bo  expressed  as  follows  : 

Theorem.  Wh(  n  a  quadratic  equation  is  reduced 
to  the  general  form 

x^  ^  px  -\-  q  —  0, 

the  coefficient  of  x  will  be  equal  to  the  sum  of  the  roots 
with  the  sign  changed. 

The  term  independent  of  x  will  be  equal  to  the 
product  of  the  roots. 

The  student  may  ask  why  can  we  not  determine  the  roots  of  the 
quadratic  equation  from  equations  (5),  regarding  a  and  j3  as  the  unknown 
quantities? 


IE 


QUAD  RATIO  EQUATIONS. 


187 


We  can  do  so,  but  let  us  see  what  the  result  will  be.    We  elimiuate 
either  «  or  (3  by  substiintion  or  by  comparison. 

From  the  second  equation  (5)  we  have, 


,  '-  ( 


Substituting  this  in  the  first  equation,  we  have 

P- 
Clearing  of  fractions  and  transposing, 


„  +  ?  = 

a 


«-  -f  pa  +  ^  =  0. 

We  have  now  the  same  equation  with  which  we  started,  only  « takes 
the  place  of  x.  If  we  had  eliminated  a,  we  should  have  had  the  same 
equation  in  j3,  namely. 

So  the  equations  (5),  when  we  try  to  solve  them,  only  lead  us  to  the 
original  equation. 

199.  To  form  a  Quadratic  Equation  when  the  Roots  are 
given.  The  foregoing  principles  will  enable  us  to  form  a  quad- 
ratic equation  which  shall  have  any  given  roots.  We  have 
only  to  substitute  the  roots  for  «  and  /3  in  equation  (4),  and 
perform  the  multiplications. 

EXERCISE  fi. 

Form  equations  of  which  the  roots  shall  be : 
I.     +1  and  —  1.  2.     3  and  2. 

_  3  and  -  2.  4-     3  +  2\/l0  and  3-2\/l0. 

7  +  21/3  and  7— 2V3.     6.     +1  and  +  2. 

—  1  and  +2.  8.     —  1  and  —  2. 

-f  1  and  —  2.  10.     2  +  Vs  and  2  —  VS. 


3- 

5. 
7. 


II. 

IS- 
17- 


3  A     4 

7  and  -• 
4  5 


12. 


'^  A^ 

2  ^^^  2" 


2+1/2  and  2-a/2.      1+     9  +  ^^2  and  9  -  2\/2 
5  +  7^5  and  5  —  7^/5.    16.     a  +  5  and  a  —  h. 
a  +  V«^  —  y^  and  a  —  ^/a^  —  b^. 


I 


f  i 


•  I 


'  i 


i^- 


■  il 


1 


188 


IMAGINARY  ROOTS. 


A 


!■ 


Equations  having:  Imaginary  Roots. 

200.  When  wo  complete  tlio  square  in  order  to  solve  a 
quadratic  equation,  the  quantity  on  the  right-hand  side  of  the 
equation  to  which  that  squ  ire  is  equal  must  be  positive,  else 
there  can  be  no  real  root.  For  if  we  square  either  a  positive 
or  negative  quantity,  the  result  will  be  positive.  Hence,  if 
the  square  of  the  first  member  comes  out  equal  to  a  negative! 
quantity,  there  is  no  answer,  either  positive  or  negative,  which 
will  fulfil  the  conditions.  Such  a  result  shows  that  impossible 
conditions  have  been  introduced  into  the  problem. 

EXAMPLES. 

1.  To  divide  the  number  10  into  two  such  parts  that  their 
product  shall  be  34. 

If  we  proceed  with  this  equation  in  the  usual  way,  we  shall 
have,  on  completing  the  square, 

xi  —  lOx  -f-  25  =  -  9, 
or  {x  —  5)2  —  —  9. 

The  square  being  negative,  there  is  no  answer.  On  con- 
sidering the  question,  we  shall  see  that  the  greatest  possible 
product  which  the  two  pavts  of  10  can  have  is  when  they  are 
each  5.  It  is  therefore  impossible  to  divide  the  number  10 
into  two  parts  of  which  the  product  shall  be  more  than  25  ;  and 
because  the  question  supposes  the  product  to  be  34,  it  is  im- 
possible in  ordinary  numbers. 

2.  Suppose  a  person  to  travel  on  the  surface  of  the  earth  to 
aiiy  distance ;  ho  tV  far  must  he  go  in  order  that  the  straight 
line  through  the  round  earth  from  the  point  whence  he  started 
to  the  point  at  whi)h  he  arrives  shall  be  8000  miles? 

It  io  evident  that  the  gre?test  possible  length  of  this  line  is 
a  diameter  of  the  earth,  namdy,  7,912  miles.  Hence  he  can 
never  get  8,000  miles  away,  and  the  answer  is  impossible. 

In  such  cases  the  square  root  of  the  negative  quantity  is 
considered  to  be  part  of  a  root  of  the  equation,  and  because  it 
is  not  equal  to  any  positive  or  negative  algebraic  quantity,  it 
is  called  an  imaginary  root.  The  theory  of  such  roots  will  be 
explained  in  a  subsequent  book. 


RE 


in 


W( 


(^ 


a 


IRRATIONAL   EQUATIONS. 


189 


>lvo  a 
)f  llie 
\  (-'l.se 
|)sitivo 
hco,  if 
Igativo 
wliich 
)ssib]e 


their 


shall 


im- 


CHAPTER    III. 

REDUCTION    OF    IRRATIONAL    E(  UATIONS    TO    THE 

NORMAL    FORM. 

201.  An  Irrational  Equation  is  one  in  which  the 
unknown  quantity  appears  under  the  radical  sign. 

An  irrational  equation  may  be  cleared  of  fractions 
in  tlie  same  way  as  if  it  were  rational. 

Example.    Clear  from  fractions  the  equation 
Vx  +  a  +  \/x  —  a  2a 


V^  +  a  —  ^/  X 


a 


Vx^ 


a" 


Multiplying  both  members  by  \/^^  —  «^  =  V^-f  a  \/x—a, 
we  have 

(x  +  a)  Vx  —  a  +  {x  —  a)  ^/x  ■\-  a  _ 

^/x  -\-  a  —  ^/x  —  a 
Next,  multiplying  by  V^  4-  «  —  V^  —  «,  wc  have 

(.^•  +  rt)  ^/x  —  a  4-  {x  —  a)  ^/x  +  «  =  lay/x^a  —  2flV'.?;  —  a. 
Transposing  and  reducing,  we  have 

{x  +  3«)  ^/x^a  +  {x  —  3a)  V^M^  =  0, 
and  the  equation  is  cleared  of  denominators. 

Clearing"  of  Surds. 

203.  In  order  that  an  irrational  equation  may  be  solved, 
it  must  also  be  cleared  of  surds  which  contain  the  unknown 
quantity.  In  showing  how  this  is  done,  we  shall  suppose  the 
equation  to  be  cleared  of  denominators,  and  to  be  composed  of 
terms  so*ne  or  all  of  which  are  multiplied  by  the  square  roots 
of  given  functions  of  x. 

Let  us  take,  as  a  first  example,  the  equation  just  found. 
Since  a  surd  may  be  either  positive  or  negative,  the  equation 
in  question  may  mean  any  one  of  the  following  four : 


I        i- 


I? 


n 


ii' 


■■  \  ( 


190 


IRRA  TIONA  L   EQ  UA  T10N8. 


[x  -\-  '6a)  ^Tc^a  +  (.?;  —  3r/)  ^Jx  +  a  =:  0, 
(./;  +  3«)  ^/x  —  a  —  {x  —  6a)  ^/x  +  «  =  0, 
{x.  +  6a)  Vx^a  +  {x  —  6a)  \/^+7i  =  0, 


(1) 

(■•5) 
(•i) 


—  (ic  +  6a)  V-^  —  a  —  {x  —  Sr*?)  V^  H-  «  =  0. 

But  the  tliird  equation  is  merely  the  negative  of  the  second, 
ai)d  the  fourth  the  negative  of  the  first,  so  tliat  only  two  have 
dilfercnt  roots.     Let  us  put,  for  brevity, 

F  =  (x  +  3a)  V^  —  a  +  {x  —  3a)  y/x  +  a, 
<2  =  ('^  +  3a)  ^/x  —  a—  (x  —  3a)  's/x  +  a, 
and  lot  us  consider  the  equation, 

pq  =  0. 

Since  this  equation  is  satisfied  when,  and  only  when,  we 
have  either  P  =:  0  or  (>  =  0,  it  follows  that  every  value  of  .6- 
whiel)  satisfies  either  of  the  equations  (1)  or  (3)  will  satisfy  (G). 
Also,  every  root  of  (G)  must  be  a  root  either  of  (1)  or  (2). 

If  we  substitute  in  (6)  the  values  of  P  and  Q  in  (5),  we 
shall  then  have 

{x  +  6aY  {x  —  a)  —  (x  —  6af  {x  -^  a)  =  0, 

which  reduces  to  hx^  —  9«2  —  0, 

3a 


(5) 


(G) 


and  gives 


x=  ± 


\/5 


It  will  be  remarked  that  the  process  by  which  we  free  the 
equation  from  surds  is  similar  to  that  for  rationalizing  the 
terms  of  a  fraction  employed  in  §  185. 

As  a  second  example,  let  us  take  the  equation. 


Va;  -{- 11  +  V^  —  4  —  5  =  0.  (a) 

Wc  Avrite  the  three  additional  equations  formed  by  combin- 
ing the  positive  and  negative  values  of  the  surds  in  every  way: 

—  ^/x  +  11  -f  ^/^ 


4  -  5  =  0, 

V^+  11  —  A/.r  — 4  —  5  =  0, 
—  \'x  +  11  —  ^/x  —  \.  —  5  =  0. 
The  product  of  the  first  two  equations  is 


.-,;! 


I  tlii^^- 


IRRATIONAL   EQUATIONS. 


191 


0) 

CO 

(i) 


(V^i 


5)2  _  (,;  +   11)    =    0, 


(5) 


0^  10  -  10\/a;^^  =  0.  (1) 

The  product  of  the  last  two  is 

10  +  loVi^^^i  =  0.  (^) 

The  product  of  these  two  products  is 

100  —  100  {x-4:)  =  0, 

which  gives  ^  =  ^' 

It  will  be  remarked  that  (2)  differs  from  (1)  only  in  having 
the  sign  of  the  surd  different.  This  must  be  the  case,  because 
tlic  second  pair  of  equations  formed  from  («)  djlTer  from  the 
first  pair  only  in  having  the  sign  of  the  surd  Vx  -  4  different. 
ITcnee  it  is  not  necessary  to  write  more  than  one  pair  of  the 
equations  at  each  step.     The  general  process  is  as  follows : 

I.  Change  the  si^n  of  one  of  the  surds  in  the  iivcn 
equation,  and  imtUiply  the  equation  thus  formed  by  the 
original  equation. 

XL  Reduce  this  product,  in  it  ehange  the  sign  of  nn- 
other  of  the  surds,  and  form  a  new  product  of  the  tiro 
equations  thus  formed.  ^ 

III.  Continue  the  process  until  an  equation  without 

surds  is  reached. 
Example.     Solve 

y'8'^T^  4-  V'^x.  +  6  -f  Vx~^  =  0. 


Changing  the  sign  of  Vx  +  4, 

The  product  is 

{Vh^T^  +  V2iT6)'  -  (.^•  +  4)  =  0, 
or,  after  reduction, 

9a:  +  11  +  2\/8^+^  V'^o:  +  6  =  0. 
Changing  the  sign  of  V'ixVQ,  we  have 

9x  q-  11  -  2V«^+9  V^x-{-  G  =  0. 


^' 


,,  ■'< 


■  > ', 


I  i 


ii 


ll 


■'I  4i 


.''■I 


f 


1 


'^    I: 


(1, 


192 


IliliA  TIONA  L    EQ  UA  TIONS. 


Tlie  product  of  the  last  two  equations  reduces  to 
I'^x^  _  66a;  -  95  =  0, 

33  ±  52 


which  being  solved  gives 


X  = 


17 


Remark.  Equations  containing  surds  may  often  reduce  to  the  form 
treated  in  §  196.  In  this  case,  the  methods  of  that  section  may  be  fol- 
lowed. 

EXERCISES. 

Solve  the  equations: 

1  1  2'v/«-2V.^ 


I. 


2. 


Vx;  +  V(i       Vx  —  V 


a 


X  —  a 


Vx'-\-a       X 

—  ■ 

a 


3.     a/o;  +  3  —  a/^c  —  4  =  1, 


4.  V'>'  +  i-i  4-  \/.<'"^T4  =  14. 

5.  (3_a;)5_(;3  +  :^;2)1  -  0. 

6.  V^  +  -v/'^'  +  Va  —  ^/x  =  'ZV^  +  -7 


7. 

8. 


+ 


V; 


X 


Vx-j-'Z      ^'  —  4       Vx  —  2 

5ar  —  9  _  ^  _  V'^'  —  3^ 
\/5^  +  3  ~     2 


=  0. 


).    V(^^  —  ^x  + 


a; 


b. 


10. 


II. 


X  +  V^;  _  x{x  —  1) 

a;  —  Va;  ^ 

a/i  +  «  _       1 

V^  —  a  +  V^a;  —  1      V^  —  1 


L.  A. 

i- 

iil 

SIMULTANEOUS   QUADRA  TIC   EQUATIONS.         193 


CHAPTER    IV. 

SIMULTANEOUS    QUADRATIC    EQUATIONS. 

Between  ji  pair  of  simulbineous  general  quadratic  equations 
one  of  tlie  unknown  quantities  can  always  be  eliminated.  The 
resulting  equation,  when  reduced,  will  be  of  the  fourth  degree 
Avith  respect  to  the  other  unknown  quantity,  and  cannot  bo 
solved  like  a  quadratic  equation. 

But  there  are  several  cases  in  which  a  solution  of  two  equa- 
tions, one  of  which  is  of  the  second  or  some  higher  degree, 
may  be  eifected,  owing  to  some  of  the  terms  being  wanting  in 
one  or  both  equations. 

203.  Case  I.  When  one  of  the  equations  is  of 
the  first  deffree  onlt/. 

This  case  may  be  solved  thus  : 

Rule.  Find  the  vahte  of  one  of  tJie  unJcnoicn  quan- 
tities i'l  terms  of  the  other  from  the  equation  of  the  first 
degree.  TJus  value  being  suhstituted  in  the  other  equa- 
tion, we  shall  have  a  quadratie  equation  from  which  the 
other  unhnown  quantity  may  he  found. 

Example.    Solve 


=1:1 


27?  +  ^xy  —  5if  —  X  —  by  —  2G, 

2x  —  'Sy 

From  the  second  equation  we  find 

3.V  +  5 


x 


Whence, 


X 


2  — 


2 

0?/2  4.  30^^/  +  25 


(^) 


if>) 


Substituting  this  value  in  the  first  equation  and  reducing, 

we  find 

4if  +  IQy  +  10  =  26. 

Solving  this  quadratic  equation, 
13 


\'' 


iM ' 


:    t 


H! 


,..!      ! 


1 


i- 


'h 


194        SIMULTANEOUS  QUADRATIC  EQUATIONS. 

y  =  —2±VS  =  -    2±  2\/2. 
This  value  of  y  being  substituted  in  the  e({uation  [b)  gives, 
—  1  ±  3\/8        —  1  ±  Oa/2 


X  = 


2 


J3 


Tlie  same  problem  mny  be  solved  in  the  reverse  order  by  eliminating 
ii  instead  of  x.    The  second  equation  (a)  gives 


y 


2x 


5 


3 


If  we  substitute  tliis  value  of?/  in  the  first  equation,  we  shall  have  a 
quadratic  equation  in  x.,  from  which  the  value  of  the  latter  quantity  can 
tje  found. 

EXERCISES. 


Solve 

I. 

x^  —  2xy  +  4y2  =  21. 
2x  +  y  —  12. 

2. 

3^  _  2^2  ^  5.^.  _  2//  —  28. 

a;  +  ?/  +  4  —  0. 

3. 

^xy  +  7^2  _  ,,  -y  _n, 
X  -\-  2y  ~  0. 

4- 

32;2  +  2y2  -  813, 
7a;      %  =    17. 

5- 

x-^y  =  1, 

X      y        7 
11      X       12 

304.  Case  II.  When  each  equation  contains 
only  one  term  of  the  second  degree^  and  that  term 
^uis  the  same  2>voduct  or  square  of  the  tinknoivn 
quantities  in  the  two  equations. 

Such  equations  are 

ax^  -f  dx  -\.  ey  -{- f  z=  0,  ] 
a'x^  +  d'x  +  e'y  +  /'  ^  0,  f 

where  the  only  term  of  the  second  degree  is  that  in  x\ 

If  we  eliminate  x^  from  these  equations  by  multiplying  the 
first  by  a'  and  the  second  by  a,  and  subtracting,  we  have 


(a) 


SIMULTANEOUS  QUADRATIC  EQUATIONS.        19.1 


{ad  —  ad')  x  +  {i('c  —  ae')  >j  +  a'/—  af  —  0. 

Solving  this  equation  with  respect  to  x^  wo  lind 

{ae'  —  a'e)  y  +  (if  —  a'f 
a'd  —  ad' 


X  = 


(*) 


By  substituting  this  value  of  x  in  either  of  the  equations 
{(i),  we  shall  have  a  (quadratic  ecjuation  in  ?/.  Solving  the 
latter,  we  shall  o1)tain  two  values  of  y.  Substituting  tliese  in 
{b),  we  shall  have  the  two  corresponding  values  of  x,  and  the 
solution  will  be  complete.     Hence  the  rule, 

.'liminate  the  term  of  the  second  decree  hi/  addition 
or  suhtractioii,  and  use  the  resulting  eqitation  of  the  first 
decree  with  either  of  the  original  equations^  as  in  Case  1. 

Example.     Solve 

2xy  —  4:r  -f  5?/  —  23,  ) 
^xy  +  7.^-  +    ^  =  41.  f 

Multiplying  the  first  equation  by  3  and  the  second  by  3, 
and  subtracting,  we  have 

—  2Qx  +  13//  =  —  13  ;  (h) 


(a) 


wlieuce, 


1 


1 


•'^  ="  2^  +  3 


W 


Substituting  this  value  in  the  first  equation,   we  find  a 
quadratic  equation,  which,  being  solved,  gives 

?/  ==  -  2  ±  a/20. 

Substituting  these  values  in  {c),  the  result  is 


X 


=  -|±iV30. 


The   two   sets  of  values   of  the   unknown  quantities  are 
therefore 


X,  =  -^  +  1^29, 

y^  r=  -  2  +  V29, 


^29. 


^8   = 


—    _  0 


We  might  ha\'e  obtained  the  same  result  by  solving  the  equation  (c) 
with  respect  to  y,  and  substituting  in  (a).  The  student  should  practice 
both  methods. 


/   . » 


;  ! 


\  I, 


I 


100        HIMULTANEOUH  qUADUAlW  J'jqrATIONS. 


U; 


J  i' 


EXERCISES. 

I.  Ca;2  —  ^x  —  Ay  =  25, 

xi  -f  ^Zx  —  lif/  z=z  18. 

a.  2//2  +  2/  =  28, 

y2  _^  3^.  —  4^  =z  18. 

3.  xfj  +  (5:c  +  7//  =  GG, 

3;ry  -\-  2x  -^  67/  =  70. 

305.  Case  III.  When  neither  equation  coU' 
tains  a  term  of  the  first  degree  in  x  or  y. 

Rule.  EliDiiiiatG  the  constant  terms  by  niultipliji'i^ 
each  c(/ nation  by  the  constant  term  of  the  other,  (did 
adding  or  suhtraeting  the  two  products.  Tlie  result  will 
be  a,  quadratic  equation,  from  whicJi  either  unJamirn 
quan,tity  can  be  determined  in  terms  of  the  other.  Then 
substitute  as  in  Case  I. 


Example.     Solve 

14xlsi:  eq., 

5  x2d  cq., 

Subtracting, 


X?  -f    ^y 


y 


2  — 


5, 


^x?  —  3.7;y  +  27/^  =  14. 
14:^2  ^_  i4^y  _  uf  —  70. 
10j;2  —  Ibxy  +  10//2  =  70. 


(1) 


4.^•2  +  29.?'?/  —  Ua/  =    0. 

This  is  a  quadratic  equation,  by  which  one  unknown  quan- 
tity can  be  expressed  in  terms  of  the  other  without  the  latter 
being  under  the  radical  sign. 

Transposing,  4:X^  +  29xy  =  24:y\ 


(2) 


841 
Completing  square,   4:X^  +  29a;^  +  ^-^^  = 


1225 
16 


r 


Extracting  root. 


Whence, 


X  = 


29  ,  35 

^^  +  'jy  =  ±'^y' 


29  ±  35 


g      y  =  ^y  o^'  -  %• 

Substituting  the  first  of  these  values  of  x  in  either  of  the 
original  equations,  we  shall  have 

2/2  =  16; 


HTMULTANEOUS   QUADHATIC   EQl/ATIONS.        107 

whence,  y  =  ±  4 ;        x=z  ±:]. 

Substituting  the  second  value  of  x,  we  have 

I 

^     -    11' 

Therefore  the  four  possible  vuUies  of  the  unknown  (luanti- 
ties  arc,  g 

x=  +'S,     -3,      +^^,     -^- 

1  ^     1_ 

Etich  of  these  four  puirs  of  vahies  satisties  tlie  original 
equation. 

A  slio-ht  change  in  the  mode  of  proceeding  is  to  divide  the 
equation"  (2)  by  either  x^  or  y^  and  to  tind  the  value  of  tho 
(,notient.     Dividing  by  y^  and  putting 

X 

y 

the  equation  will  become 

4?*2  +  20^6  —  24  =  0. 

This  quadratic  equation,  being  solved,  gives 

_  29  ±  35  _  3  g 


1 


».i 


'    il 


I    :  #■ 


4 


!'  i-i 


Putting  -  for  it,  and  multiplying  by  y, 
X  =  -y  or  ~Sy,  as  before. 


Solve 


I. 


2. 


EXERCISES. 


^^2  _    a-y  +    2/2 
a;2  _  2xy  +  42/2 

-3 

—  4 

=  0, 
-  0. 

22:2  4.  ^xy  — 
0?  +  3^//  - 

47/2 

—  2 
+  1 

=  0, 
-  0. 

.;  I 


f 

I  .[; 


'ii : 

>  il 


'  "  I 


!; 


s. 


198         SIMULTANEOUS   QUADIiATIC    EQUATIONS. 

200.  C.vsi:  IV.  When  fho  crprcssfons  rontnht'- 
inn  '''''  *(n/>uoirH  qunntUies  in  the  tiro  eqiidfions 
hdvr  coiHitioii  /'actors, 

UiLi].  Diride  one  of  the  rqitofious  which  can  he  fac- 
tnrcd  1)1/  the-  other,  and  cancel  the  coninion,  factor:^. 
Then  ctear  of  frnction,^,  if  necessary,  and  ice  shall  have 
an  equation  of  a  lower  deo'ree, 

EXAMPLES. 

1.  a^^  +  /  =  01,    ./•  +  v/=  7. 

Wo  have  seen  (§  94,  Th.  1)  tlifit  cfi-\-y^  is  divisible  by  x-{-}/. 
So  dividing  the  first  C([ii{ition  by  tbe  second,  we  have 

a;8  _  X1J  4-  .y^  =  13. 
This  is  an  equation  of  the  second  aepfrcc  only,  and  when 
combined  with  tlie  second  of  the  original  cf|iiations,  the  solu- 
tion may  be  effected  by  Case  I.     The  result  is, 
X  =  3  or  4,        y  =  4  or  3. 

2.  xtf  4-  !l^  =  133,    .1-2  —  ?/2  =  95. 

Factoring  the  first  member  of  each  equation,  the  equations 
become 

y  {x  +  y)  =  133,        {x  +  y)  (x  -  y)  =  95. 

Dividing  one  equation  by  the  other,  and  clearing  of  fractions, 


IV 


12?/  =  7.r,    or    y  =  ~^x. 

The  problem  is  now  reduced  to  Case  I,  tliis  value  of  y 
being  combined  with  either  of  the  original  equations. 

207.  There  are  many  other  devices  by  which  simultaneous 
equations  may  be  solved  or  brought  under  one  of  the  above 
cases,  for  which  no  general  rule  can  be  given,  and  in  which 
the  solution  must  be  left  to  the  ingenuity  of  the  student. 
Sometimes,  also,  an  equation  which  comes  under  one  of  the 
cases  can  be  solved  much  more  expeditiously  than  by  the  rule. 

Let  us  take,  for  instance,  the  equations, 

x^  -\-  y^  =  G5,        xy  =  28. 

These  equations  can  be  solved  by  Case  III,  but  the  work 
would  be  long  and  cumbrous.     We  see  that  by  adding  and 


BUiUL  TA  SIuO  US   q  UA  DUA  Tld   KQ  U.  1  Tioys.         190 

Bubtmcting  twice  tlio  second  cMjiiation  to  and  from  tlic  fii-Hl, 
ue  am  lorin  two  pcrloct  Hiiuans.  Kxtrjuliiig  tlu!  nxtts  of 
tlu'Si'  s(iUiiiTS,  wc  shall  liuve  two  siinplc  c'(iMatioiis,  which  sliall 
i-ive  the  sohitiou  at  once.  Eacli  unknown  <|uantitv  will  have 
four  value.><,  namely,   ±  T  ±  *1. 

PF<ODLEMS    AND     EXERCISES. 

Tho  foUowiii'j^  eiiuatioiirt  cJin  nil  bo  solved  l)y  hoiuo  nhort  and  cxpf. 
ditious  combiuution  of  the  ci|Uutioii8,  or  by  fuclonng,  without  ^oiii^ 
through  thti  coiupU'X  procoas  of  Case  HI.  Tho  student  is  rccomimndcd 
not  to  work  upon  the  (-(lUtttioiiH  at  raiidoni,  but  to  study  enc'li  pair  until 
he  sees  how  it  can  bo  reduced  to  a  simpler  ecjuation  by  addition,  multi- 
plication, or  factoring,  and  then  to  go  through  tho  operations  thu«  BUg- 
gested. 

1.  y"^  -\-  xy  =  14,     x^  -\-  ory  =  35. 

2.  4a?2  —  2xy  =  208,     2.ry  —  y/2  =r  39. 

3.  x^  +  y  =  '\x,     ?/2  +  J-  =  4y. 

If  we  subtract  ono  of  these  e(i nations  from  tho  other,  the  difTerenco 
will  be  divisible  by  x  —  y. 

4.  .^•3  4-  Z/''  +  3.r  -|-  3//  =  378,    x^  +  y/3  _  3^  _  3^  _  324. 

5.  x^  +  ?/2  =  74,    x^y  zzz  12. 

6.  x^  -\-  xy  =  G3,    x^ 

Vx  +  Vy 


4, 


i/2  =  77. 

X2  —  7/2  _  544^ 


8. 

9- 
10. 

II. 

12. 

14. 
15. 


5. 


Vx  —  Vy 

x^  +  xy  =.  a,    2/2  .j_  ^y  z=  b. 

ai^  4-  xy^  =10,     ?/^  +  x'^y 

X  =  aV^v  -\-  y,     y  =  hVx  +  y. 

xVx  +  ?/  =  12.     yV'x  -\-  y  =  15. 

2x2  _|_  2y^  =  X  -\-  y,     x^  i-  y'^  =  X  —  y. 

63^  —  5?/2  =z  X  ■]-  y,     3a;2  —  3^/2  =  x  —  y. 

x^  +  y^-\-z^  z='dO,     xy-j-yz-{-zx  =  17,    x  —  y  —  z  =  2. 


\Jx- 


x-\-y 


6;?/ 


y 


-^xr-^-y-  = 


Gy 


2, 


-Vi- 


y 

y 


8 


^  —  2/ 


(■ .  „ 


(    I 


I     ■  ll 
1     i 


.  ■■    i 

; 
1 

^"■': 

hi ' 

-.iU 

200 


PUOBLEMc 


t  .' 


i6.  A  principal  of  65000  amounts,  with  simple  interest,  to 
$7100  after  a  certain  number  of  years.  Had  tlie  rate  of  inter- 
est been  1  i)ercent.  bigherand  the  time  1  year  longer,  it  would 
have  amounted  to  -^7800.     What  was  tlie  time  and  rate? 

17.  A  courier  left  a  station  riding  at  a  uniform  rate.  Five 
hours  afterward,  a  second  followed  him,  riding  3  miles  an 
hour  fast(  r.  Two  liours  after  the  second,  a  third  started  at 
the  rate  of  10  miles  an  hour.  They  all  reach  tlieir  destination 
at  the  same  time.     Wiiat  was  its  distance  and  the  rate  of  riding  ? 

18.  In  a  right-angled  triangle  there  is  given  the  hypothe- 
nuse  =  (If  and  the  area  =  h^',  find  the  sides. 

19.  Find  two  numbers  such  that  their  product,  sum,  and 
difiereuce  of  squares  shall  be  equal  to  each  other. 

20.  Find  two  numbers  whose  product  is  21 G ;  and  if  tlio 
greater  be  diminished  by  4,  and  the  less  increased  by  3,  the 
product  of  this  sum  and  difference  may  be  2-40. 

21.  There  are  two  numbers  whose  sum  is  7-4,  and  the  sum 
of  their  square  roots  is  12.     What  are  tho  numbers  ? 

22.  Find  two  numbers  whose  sum  is  72,  and  the  sum  of 
their  cube  roots  0. 

23.  The  sides  of  a  given  rectangle  are  m  and  n.  Find  the 
sides  of  another  wliich  shall  have  twice  the  perimeter  and  twice 
the  area  of  the  given  one. 

24.  A  certain  number  of  workmen  require  3  days  to  com- 
plete a  work.  A  number  4  less,  working  3  hours  less  per  day, 
will  do  it  in  G  days.  A  number  G  greater  than  the  original 
number,  working  6  hours  less  per  day,  will  complete  the  work 
in  4  days.  What  was  the  original  number  of  workmen,  and 
how  long  did  they  work  per  day  ? 

25.  Find  two  numbers  whose  sum  is  18  and  the  sum  of 
their  fourth  powers  14096. 

Note.  Since  the  sum  of  the  two  numbers  is  18,  it  is  evident  that 
the  one  must  be  as  much  less  than  9  as  the  other  is  greater.  Tlie  equa- 
tions will  assume  the  simplest  form  when  we  take,  as  the  unknown  quan- 
tity, the  common  amount  by  which  the  numbers  differ  from  9 

26.  Find  two  numbers,  x  and  y,  such  that 

a;3  _|_  2/3  :  ^  __  2/3    : :    35  :  19, 

xy  =  24. 

27.  Find  two  numbers  whose  sum  is  14  and  the  sum  of 
their  fifth  powers  1G1204. 


BOOK    VII. 
PROGRIi  SSIOiVS. 

CHAPTER     I. 
ARITHMETICAL     PROGRESSION. 

308.  Def.  When  we  have  a  series  of  numbers  each 
of  which  is  greater  or  less  than  the  preceding  by  a  con- 
stant quantity,  the  series  is  said  to  form  an  Arithmet- 
ical Progression. 

Example.     The  series 


^h,    etc., 


7,  12,  17,  22,  27,  32,  etc. 
7,  5,  3,  1,  -1,  -3,  etc. 
a  -\-  b,     a,     a  —  h,     a  —  2b,     a  - 

are  each  in  arithmetical  progression,  because,  in  the  first,  each 
number  is  greater  than  the  preceding  hy  5 ;  in  the  second, 
each  is  less  than  the  preceding  by  2 ;  in  the  third,  each  is  less 
than  the  preceding  by  b. 

Def.  The  amount  by  which  each  term  of  an  aritli- 
metical  progression  is  greater  tlian  the  preceding  one  is 
called  the  Common  Difference. 

Def.  The  Arithmetical  Mean  of  two  quantities  is 
lialf  their  sum. 

All  the  terms  of  an  arithmetical  progression  except 
the  iirst  and  last  are  called  so  many  arithmetical  means 
between  the  first  and  last  as  extremes. 

Example.  The  four  numbers.  5,  8,  11, 14,  form  the  four 
arithmetical  means  between  2  and  17. 


;  f 


M 


I- 


L  •  I 


,  f 


:] 


jii 


f1  !' 

'  I'  u 


•'I 

,1    h 
,'!  n 

t  jt    i-j 


1    ,'''i* 


202 


ABITUMETICAL   PHOGRESSION. 


r^ 


is 


si 


EXERCISES. 

1.  Form  four  terms   of   the   aritlimotical   progression   of 
wliich  the  first  term  is  7  and  common  difference  3. 

2.  Write  tlie  first  seven  terras  of  the  i)rogression  of  wliieli 
tlie  first  term  is  11  and  tlie  common  difference  —  3. 

3.  Write  five  terms  of  the  progression  of  wiiich  tlie  first 
term  is  a  —  ^n  and  the  common  dilference  'Zn. 

Problems  in  Progr^ession. 

309.  Let  us  put 

u,  the  first  term  of  a  progression. 
d,  the  common  difference. 
Uy  the  number  of  terms. 
I,  tlie  last  term. 
2,  the  sum  of  all  the  terms. 
The  series  is  then 

a,    a-\-d,    a -{-2d,   .  .  .  .  I. 

Any  three  of  the  above  five  quantities  being  given,  the 
other  t;vo  may  be  found. 

Pkoblem  I.  Given  the  first  term,  the  coimnon  differ- 
ence, tutd  the  nmnher  of  terms,  to  find  tlie  last  term. 

The  1st  term  is  here  cf, 

2d      "        "  a  +  d, 

3d      "        "  a-\-  2d. 

The  coeffiGicnt  of  d  is,  in  each  case,  1  less  than  the  number 
of  the  term.  Since  this  coefficient  increases  by  unity  for  every 
term  wo  add,  it  must  remain  less  by  unity  than  the  number  of 
the  term.     Hence, 

The  i^'^  term  is  .'>  f  {i  —  1)  d, 
whatever  be  i.     Hence,  when  i  =  n, 

I  ~  a  Jr  (n  —  1)  d.  (1) 

From  this  equation  w^e  can  solve  the  further  problems : 

Problem  II.  Given  the  last  term  I,  the  common  dif- 
ference d,  ctnd  the  number  of  terms  n,  to  find  the  first 
term. 


AlUTHMETIGAL   PROQRESSION. 


203 


The  solution  is  found  by  solving  (1)  with  respect  to  a, 

vbich  gives 

a=:l-{n-  1)  d.  C-i) 

Problem  III.  Given  the  first  and  last  terms,  a  and  I, 
and  the  number  of  terms  n,  to  find  the  common  differ- 
ence. 

Solution  from  (1),  d  being  tlie  unknown  quantity, 

cl  =  tzl.  (3) 

Problem  IV.  Given  tlie  first  and  last  terms  and  the 
common  difference,  to  find  the  number  of  terms. 

Solution,  also  from  (1), 


I 


»  =  -^-  +  1  =  ^ (4) 

Problem  V.     To  find  the  sum  of  all  the  terms  of  an 
(irithmetical  progression. 

We  have,  by  the  definition  of  2, 

1.  —  a  -\-  {a  +  d)  -\~  {a  -\-  '2d)  + {I  —  d)  ■}-  I, 

tlie  parentheses  being  used  only  to  distinguish  the  terms. 

Now  let  us  write  the  terms  in  reverse  order.     The  term 
Ijcfore  the  last  is  I  —  d,  the  second  one  before  it  /  —  M,  etc. 

We  therefore  have, 

S  —  ?  +  (^  —  f/)  4-  (/  —  2r7) +  {a  +  ^0  +  a. 

Adding  these  two  values  of  2  together,  term  by  term,  we 
find 
2v  =,  {a  +  l)  +  (rt  +  0  +  (rt  +  0  + +  («  +  /)  +  («  +  /), 

the  quantity  {n^l)  being  written  as  often  as  there  are  terms, 
that  is,  n  times.     Hence, 

25:  =  n  {a  +  0^ 
a  +  I 


n 


(5) 


a  +  I 


Remark.    The  expression  — J-,  that  is,  half  the  sum  of 
the  extreme  terms,  is  the  mean  value  of  all  the  terms.     The 


.  *i 


if 


» J!    i 


I   'I    !1 

n 


'  11 

;    I 


i\ 


I 


\  i' 


.)  ': 


!ti 


'^ 


204 


A  UITUMETICAL    PliOQREHSION. 


sum  of  the  n  terms  is  therefore  the  same  as  if  each  of  them 
had  this  value. 

*^10.  Ill  the  equation  (5)  we  are  sujjposed  to  know  tlic; 
first  and  last  terms  and  the  numher  of  terms.  If  other  quan- 
tities are  taken  as  the  known  ones,  we  have  to  substitute  lor 
some  one  of  tlie  quantities  in  (5)  its  expression  in  one  of  the 
e([uations  (1),  (2),  (Ij),  or  (4).  Suppose,  for  example,  that  wu 
have  given  only  the  last  term,  the  common  difference,  and  the 
number  of  terms,  that  is,  /,  d,  and  n.  We  must  then  in  (5) 
substitute  for  a  its  value  in  (2).     This  will  give, 


(«) 


1 


(       n—  1    \  n{n  —  1) 

2  —  n\l :r —  a]  =  nl  —  — ^— r a. 

\  2       /  'Z 

EX  ERCISES. 

In  arithmetical  progression  there  are 

1.  Given,  common  difference,  -j-  3;  third  term  =  10. 
Find  first  term.  Ans.    First  term  =:  4. 

2.  Given  4th  term  =  b,  common  difference  =  —  c. 
Find  first  7  terms,  their  sum  and  product. 

3.  Given  3d  term  z=z  a  +  b,  4th  term  =  a  +  2b. 
Find  first  5  terms. 

4.  Given  1st  term  =  a  —  b,  9th  term  —  da  -\-  lb. 
Find  2d  term  and  common  dilference. 

5.  Given,  sum  of  0  terms  =  108. 

Find  middle  term  and  sum  of  1st  and  0th  terms. 

6.  Given  5th  term  —  7x  —  hy,  7th  term  =  0:c  —  'dy. 
Find  first  7  terms  and  common  difference. 

7.  Given  1st  term  =  12,  50th  term  =  551. 
Find  sum  of  all  50  terms. 

8.  To  find  the  sum  of  the  fiivf  100  numbers,  namely, 

1+2+3 +99  +  100. 

Here  the  first  term  a  is  1,  the  last  term  I  100,  and  the  number  of 
teiTDS  100.     The  aolution  'vj>  by  Problem  V. 

9.  Find  the  sum  of  the  first  n  entire  numbers,  namely, 

1  +  24-3....  +  n. 


AltlTlIMETlCA  L    PROGRESSION. 


205 


10.  Find  the  sum  of  tlio  lirst  n  odd  numbers,  namely, 

1  +  3  4-  5  ...  .  +  2?i  —  1. 
Here  the  number  of  tc^ruis  ib  /i. 

1 1.  Find  the  sum  of  the  first  n  even  numbers,  namely, 

2  -f-  4  +  0  .  .  .  .  +  'Zn. 

12.  In  a  school  of  m  scholars,  the  highest  recei\'ed  134 
iin'i-ii  marks,  and  each  succeeding  one  G  less  than  the  one  next 
aiiove  him.  How  many  did  the  lowest  scholar  receive?  How 
many  did  they  all  receive? 

13.  The  first  term  of  a  series  is  m,  the  last  term  2/><,  and 
the  common  difFerence  d.     What  is  the  number  of  terms? 

14.  The  first  term  is  k,  the  last  term  10^•  —  1,  and  the 
number  of  terms  0.     What  is  the  common  difference  ? 

15.  The  middle  term  of  a  progression  is  s,  the  number  of 
terms  5,  and  the  common  dilference  —  h.  What  are  the  first 
and  last  terms  and  the  sum  of  the  5  terms  ? 

16.  The  sum  of  5  numbers  in  arithmetical  pi  .gression  is 

20  and  the  sum  of  their  squares  120.     What  are  the  numbers? 

Note.  In  (questions  like  this  it  is  better  to  take  the  middle  term  for 
one  of  the  unknown  quantities.  The  other  unknown  quantity  will  be 
thi'  common  difi'iTcnce. 

17.  Find  a  number  consisting  of  three  digits  in  arithmeti- 
cal progression,  of  which  the  sum  is  15.  If  the  number  bo 
diiiiinished  by  792,  the  digits  will  be  reversed. 

18.  The  continued  product  of  three  numbers  in  aritiimet- 
ical  progression  is  640,  and  the  third  is  four  times  the  first. 
What  are  the  numbers  ? 

19.  A  traveller  has  a  journey  of  132  miles  to  perform.    lie 

goes  27  miles  the  first  day,  24  the  second,  and  so  on,  travelling 

3  miles  less  each  day  than  the  day  before.     In  how  many  days 

will  he  complete  the  journey  ? 

Here  we  have  given  the  first  term  27,  the  common  difference  —3,  and 
the  sum  of  the  terms  132.  To  solve  this,  we  take  eciuation  (5),  and  sub- 
stitute for  1  its  value  in  (1).     This  makes  (0)  reduced  to 

rt  +  rt  4-  (n  —  1)  d                  n  in  —  \)  d 
2  =  n ^' — — — -  =  na  -\ 

2,  a,  and  d  are  given  by  the  ]iroblem,  and  n  is  the  unknown  quan- 
tity. Substituting  the  numerical  value  of  the  unknown  (quantities,  the 
equation  becomes 


;  fi 


i  ■ 


1 


f 


20G 


AlirrJIMETlCAL   PliOUUEtiiilON. 


!, 


•     f: 


^ 


\ 


V-yi  =z  2'7}i  —  3 


n 


(n  -  1) 


2 


This  rcduretl  toa(iuadriitic  ccjuation  in  n,  the  solution  of  wliioh  f,'ivtH 
two  values  of  n.  Tlic  ntutl»,'iit  ishoulil  explain  this  double  answer  liv 
CDutiiiuiiit,'  the  progression  to  11  terms,  and  allowing  what  the  negative 
terms  inilicate. 

20.  Taking  the  same  question  as  tlie  last,  only  suppose  the 
distance  to  be  140  miles  instead  of  132.  JSJiow  that  the  answer 
Avill  be  imaginary,  and  explain  this  result. 

21.  A  debtor  owing  $100  arranged  to  pay  35  dollars  the 
first  month,  23  the  second,  and  so  on,  2  dollars  less  each 
month,  until  his  debt  should  l)e  discharged  How  many  puy- 
ments  must  he  make,  and  Avhat  is  the  explanation  of  the  two 
answers  ? 

2  2.  A  hogshead  holding  135  gallons  has  3  gallons  poured 
into  it  the  first  day,  G  the  second,  and  so  on,  3  gallons  more 
every  day.     How  long  before  it  will  be  filled  ? 

23.  Tlie  continued  product  of  5  consecutive  terms  is  12320 
and  their  sum  40.     AVhat  is  the  progression? 

24.  Show  that  the  condition  that  three  numbers,  p,  q,  and 
r,  are  in  arithmetical  progression  may  be  expressed  in  the  form 

'J  -p  =  - 1. 

q  —  r 

25.  In  a  progression  consisting  of  10  terms,  the  sum  of  the 
ist,  3d,  5th,  7th,  and  9th  terms  is  90,  and  the  sum  of  the  re- 
maining terms  is  110.     What  is  the  progression  ? 

26.  In  a  progression  of  an  odd  number  of  cerms  there  is 
given  the  sum  of  the  odd  terms  (the  first,  third,  fifth,  etc.)^ 
and  the  sum  of  the  even  terms  (the  second,  fourth,  etc.). 
Show  that  wo  can  find  tlie  middle  term  and  the  number  oi" 
terms,  but  not  the  common  difference. 

27.  In  a  progression  of  an  even  number  of  terms  is  given 
the  sum  of  the  even  terms  =  119,  the  sum  of  the  odd  terms  — 
105,  and  the  excess  of  the  last  term  over  tlie  first  =  2G.  Wluit 
is  the  progression  ? 

28.  Given  a  and  /,  the  first  and  last  terms,  it  is  required  to 
insert  i  arithmetical  means  between  them.  Find  the  expres- 
fcion  for  the  /  terms  required. 


IS  a 


GEOMETRICAL    PllOGRES.'^ION. 


207 


CHAPTER     II. 

GEOMETRICAL     PROGRESSION. 

*2ll.  Def.  A  Geometrical  Progression  consists  of 
,1  series  of  terms  of  which  each  is  formed  by  multiply- 
ing the  term  preceding  by  a  constant  factor. 

All  aritlimetical  progression  is  formed  by  contiiiiuil  iiddi- 
tioii  or  subtraction;  u  geometrical  progression  by  repeated 
iiiultiplieation  or  division. 

D(f.  The  factor  by  which  eacli  term  is  nuiltiplied 
to  form  the  next  one  is  called  the  Common  Ratio. 

The  common  ratio  is  analogous  to  the  common  dilference 
in  an  arithmetical  progression. 

In  other  respects  the  same  definitions  ai)ply  to  both. 

EXAMPLES. 

3,     G,     18,     54,     etc., 
is  a  progression  in  wliich  the  iirst  term  is  3  and  the  common 
ratio  3. 

3      1      1       ^      -^ 

'    '  r  4'  8' 

is  a  progression  in  which  tlie  ratio  is  -• 


etc., 


tr 


f     ! 


.,   '   I 


'<■ 


|i 


'   I 


+  3,     —  6,     4-  12,     —  24,     etc., 
is  a  progression  in  Avhich  the  ratio  is  —  2. 

Note.  A  progression  like  the  second  one  above,  formed  by  dividing 
pfidi  term  by  tlie  same  divisor  to  obtain  the  next  term,  is  included  in  tlie 
gt'iieral  definition,  because  dividing  by  any  number  is  the  same  as  multi- 
plying by  the  reciprocal.  Geometrical  progressions  may  therefore  \>x: 
divided  into  two  classes,  increasing  and  decreusing.  In  the  increasing 
progression  the  common  ratio  is  greater  tuan  1  and  the  terms  go  on  in- 
creasing ;  in  a  diminishing  progression  the  ratio  is  less  than  unity  and 
the  terras  go  on  diminishing. 

Rem.  In  a  progression  in  which  the  ratio  is  negative,  the 
terms  will  be  alternately  positive  and  negative. 


.  ii 


208 


GEOMETRIC  A  L   PR0GUE81S10N. 


Dcf.     A  Geometrical  Mean  between  two  quantities 
is  the  square  root  of  their  product. 

EXERCISES. 

Form  five  terms  of  each  of  the  following  geometrical  pro- 
gressions : 

1.  First  term,  1  ;  common  ratio,  5. 

2.  First  term,  7  ;  common  ratio,  —  3. 

3.  First  term,  1  ;  common  ratio,  —  1. 

2  3 

4.  First  term,  -  ;  common  ratio,  j- 

4  1 

5.  First  term,  -  ;  common  ratio,  ^' 


i\ 


♦     I 


Problems  of  Geometrical  Progression. 

^12.  In  a  geometrical  progression,  as  in  an  arithmetic!!  1 

one,  there  are  live  quantities,  any  three  of  which  determini' 

the  progression,  and  enable  the  other  two  to  be  found.     They 

arc 

a,    the  first  term. 

;-,  the  common  ratio. 

??,  the  number  of  terms. 

I,  the  last  term. 

2,  the  sum  of  the  71  terms. 

The  general  expression  for   the  geometrical  progression 
will  be 


a,    ar,     ar%    ar%    etc., 

because  each  of  tliese  terms  is  formed  by  multiplying  the  pic- 
ceding  one  by  r. 

The  same  problems  present  themselves  in  the  two  progro.;- 
sions.    Those  for  the  geometrical  one  are  as  follows : 

Peorlem  I.     Given  the  first  term,  the  coimnon  ratio, 
and  the  niniiher  of  terms,  to  find  the  last  term. 

The  progression  will  be 

a,     ar,     ar^,     etc. 

We  see  that  the  exponent  of  r  is  less  by  1  than  the  number 
of  the  term,  and  since  it  increases  by  1  for  each  term  added,  it 


(jmMETRICA  L    PUOOIIESSJON. 


201) 


must  remain   less   by  1,  how  many  terms  so  ever  we  take. 
Hence  the  n"'-  term  is 

1  =  r^?-«-i.  (1 ) 

Problem  II.     Given  the  last  trviti,  the  comnioti  ratio, 
and  the  nuinber  of  terms,  to  jiiid  the  first  term. 

Tiie  .sohition  is  found  by  dividing  both  members  of  (I)  by 
,.?i-i^  which  gives 

7 


a 


rtl-l 


Problem  III.    Given  the  first  term,  the  last  term,  and 
the  number  of  terms,  to  find  the  comDion  ratio. 

From  (1)  we  find 


/f>n—\  -— 


a 


Extracting  the  {n  —  1)^'*  root  of  eaeli  member,  we  have 


r 


^r- 


[The  sohition  of  Problem  IV  requires  us  to  find  n  from 
equation  (1),  and  belongs  to  a  higher  department  of  Algebra.] 

Problem  V.     To  find  the  sum  of  all  n  terms  of  a  geo- 
inetrieal  progression. 

We  have     S  :=  a  -{■  ar  -\-  ar^  -^^  etc.  -f-  «r"~^ 
Multiply  both  sides  of  this  equation  by  /'.     We  then  have 
r2  =  ar  4-  ar^  -\-  ar^  +  etc +  r/r'*. 

Now  subtract  the  first  of  these  equations  from  the  second. 
It  is  evident  that,  in  the  second  equation,  each  term  of  the 
second  member  is  equal  to  the  term  of  the  second  member  of 
the  first  equation  which  is  one  place  farther  to   the   right. 
Hence,  when  we  subtract,  all  the  terms  will  cancel  each  otiier 
except  the  first  of  the  first  equation  and  the  last  of  the  second. 
Illustration.    The  following  is  a  case  in  wliioh  a  =  2,  r  =  '6,  n  =:G: 
1  -  2  +  G  +  18  +  04+162  +  48G. 
32  =  6 +  18 +  54 +  162 +  486 -1-1458. 
Subtracting,    3S-2  =  1458  -  2  =  1456, 

or    22  =  1456,    and    2  =  728. 
14 


I 


»' 


4    I 


1  ;i' 


I  ■ 


210 


(ih:(})ii<:TiiiCA L  PiioajiKss/c tV. 


«. 


w 


Iicturning  to  flic  general  prol)lem,  we  Imvo 

(r  -  1)  1  —  rn-"  —  a   ::=z  a  (/•»  —  1)  ; 


whcneo, 


/" 


r"  —  1  1 

r  —  1  1  —  r 


(4) 


It  will  l)e  mosi convi'uicnt  to  use  the  first  form  wlieii  r  >  1, 
and  tlie  seeond  when  r  <  1. 

By  tliis  fonnulii  wo  are  enabled  to  conipnte  the  sum  of  tiie 
terms  of  a  geometrieul  jjrogression  without  autuully  forming 
all  tliu  terms  and  adding  them. 


EXERCISES 


3  lifN'f!- 


1M. 


I. 


2. 


3 
Given  '.]d  term  =  0,  eommon  ratio  =  ~. 

Find  first  5  terms. 


33  o 

--,  common  ratio  =  —  .-« 
^7  o 


Given  5th  term  = 
Find  lirst  5  terms. 

3.  Given  ath  term  r=  .r'//''',   1st  term  ==  fj\ 
Find  common  ratio. 

4.  (Jiven  1st  term  =  1,  4t]i  tenn  =;i  aK 
Find  common  ratio  and  lirst  3  terms. 

5.  Given  2d  term  =  in,  common  ratio  :=  —  711. 
Find  first  4  terms. 

6.  A  farrier  having  told  a  coachman  that  he  would  charge 
liim  $3  for  shoeing  his  horse,  the  latter  objected  to  the  prii-e. 
The  farrier  then  oflFered  to  take  1  cent  for  tlie  first  nail,  2  for 
the  second,  4  for  the  third,  and  so  on,  doubling  the  amount 
for  each  nail,  which  offer  the  coachman  accepted.  There  were 
32  nails.  Find  how  much  the  coachman  had  to  jDay  for  the 
hist  nail,  and  how  much  in  all.     (Compare  §  1G8,  Rem.) 

7.  Find  the  sum  of  11  terms  of  the  series 

2  -f-  0  +  18  +  etc., 
in  which  tlie  first  term  is  2  and  the  common  ratio  3. 

8.  Tf  the  common  ratio  of  a  progression  is  ?•,  what  will  be 
the  common  ratio  of  the  progression  formed  by  taking 

I.  Every  alternate  term  of  the  given  progression  ? 
II.   Every  71^'''  term  ? 


a  KG  METRIC.  1 A    riWC  RF.SSION, 


211 


9.  Tho  sumo  \\\\\\^  hcin;^'  supposed,  wluit  will  1)0  the  com- 
mon ratio  of  the  progression  of  w  liich  every  iiiternate  term  is 
e([iial  to  every  third  term  of  the  fj^iven  prot^ression  ? 

10.  Sliow  that  if,  in  a  geometrieai  pro<iression,  each  term 
lie  added  to  or  suhtraeted  from  tliat  next  following,  tiie  sums 
(ii-  renniinders  will  form  a  geometrical  progression. 

ir.  Show  that  if  the  arithmetical  and  geometrical  means 
of  two  (pumtitios  i30  given,  tlie  (pnmtitiis  themselves  may  be 
found,  and  give  the  expressions  for  them. 

12.  The  sum  of  the  first  and  fourth  terms  of  a  progression 
is  to  tho  sum  of  the  second  and  third  as  'ZV  \  b.  What  is  tho 
common  ratio? 

13.  Express  the  continued  product  of  all  the  terms  of  a 
geometrical  progression  in  tei-ms  o\' ti,  r,  iiiid  n? 

Limit  oftlRvSiim  of  a  Projj^ivssion. 

2Ui,  Tlieorem.  If  the  common  ratio  in  a  geonictri- 
cal  2>i'ogression  is  less  tlian  unity  (more  (\\actly,  if  it  is 
contained  between  the  limits  —1  and  +1),  then  there 
will  be  a  certain  quantity  which  the  sum  of  all  the 
terms  can  never  exceed,  no  matter  how  many  terms  we 
take. 

For  example,  the  sum  of  'he  progression 

111, 

^  +  -  +  ^  +  ete., 

in  which  the  common  ratio  is  -,  can  never  amount  to  1,  no 

matter  how  many  terms  we  take.  To  show  this,  suppose  that 
one  person  owed  another  a  dollar,  and  proceeded  to  pay  him  a 
series   of  fractions   of  a   dollar  in   geometrical   progression, 

namely, 

1111 

etc. 


1 
2' 


1 

4^ 


1 

8' 


1 
16' 


When  he  paid  him  the  ^  he  would  still  owe  another  ^, 

when  he  paid  the  t  he  would  still  owe  another  -, ,  and  so  on. 

4  4 


.  ? 


I    ; 


r 


. .  I 


Ml 


f) 


212 


GKOMHTniC.  I  /.    PliOGRKSSION. 


t 


•ki 


That  is,  at  evi-ry  i)avrntMit  lie  would  (liscliar;,^'  ono-lmlf  tlic  n  ■ 
mainiiig  tk'l)L  Muw  thcrt'  jiru  two  propusitioiis  to  bo  undii- 
stood  in  re  ft' IV  lieu  to  this  suhject. 

I.  7%P'  enfirr  <frht  ran.  never  be  (iisr/ifiro'ed  hi/  fiiirh 
prnjinefUs. 

l\)y,  since  the  dc'))t  is  halecdni  every  payment,  if  there  \v,i^ 
any  payment  wliieh  diseliar^cd  tlie  whole  reinainin<;  (U'ht,  tip 
lialf  of  a  thin^jf  would  he  ('(jual  to  the  whole  of  it,  which  is 
im])ossil)lc. 

II.  The  (h'ht  e(t\i  he  redueed  hehnr  nnij  nssi^nah/r 
limit  l)ij  rontinitin^  to  junj  Indf  of  if. 

For,  however  small  the  debt  may  he  made,  another  pay- 
ment will  make  it  smaller  by  one-half;  hence  there  is  Ud 
smallest  jimount  below  which  it  cannot  be  reduced. 

TlicH(^  two  propositioiiH,  wliicli  poem  to  oppose  oncli  othor,  hold  tlir 
triitli  Ix'twrcn  tlicm,  ns  it  woro.  Tlicy  ronptaiilly  outer  into  tlio  liiglin- 
luatlionintif's,  nnd  hlioiild  l)f'  well  undt'rstood.  W«  thereforo  prerttni 
anotlier  illustration  of  tho  same  subject. 


B 


I      I 


i 


i      tV 


Suppose  AH  to  be  a  line  of  given  length.  Let  us  go  onc- 
lialf  the  distance  from  A  to  B  at  one  step,  one-fourth  at  the 
second,  one-eiglith  at  tlie  third,  etc.  It  is  evident  that,  at  each 
step,  we  go  hi  "^  the  distance  which  remains.  Hence  the  two 
principles  judt  cited  apply  to  this  case.     That  is, 

1.  We  can  never  reach  B  by  a  series  of  such  steps,  because 
we  shall  always  have  a  distance  equal  to  the  last  step  left. 

2.  But  we  can  come  as  near  B  as  we  please,  because  every 
step  carries  us  over  half  the  remaining  distance. 

This  result  is  often  expressed  by  saying  that  we  should  reach  B  by 
taking  an  infinite  number  of  steps.  Tbis  is  a  convenient  form  of  expres- 
sion, and  we  may  sometimes  use  it,  but  it  is  not  logically  exact,  becaase 
no  conceivable  number  can  be  really  infinite.  Tbe  assumption  that  in- 
finity is  an  algebraic  quantity  often  leads  to  ambiguities  and  difficulties 
in  tbe  application  of  mathematics. 


GKOhfETIUdA  L    VUnaiiKSSlON. 


2i:^ 


ise  everv 


Def>  Tli«>  Limit  of  tlio  Huiii  1  of  a  poomptricnl 
progresHion  is  a  (jujintlty  wliicli  1  may  appioacli  so 
tliat  its  diflVn^ncc  shall  ho  Iohh  than  any  (juaiitity  we 
choose  to  assign,  but  wliich  i  can  nt'vcr  n'ach. 

EXAMPLES. 

1.  Unity  is  tliu  hmil  of  tiic  smn 

l  +  i  +  «  +  l7:  +  ^*<'- 

2.  Tlio  point  W  in  tho  procodin^  fipfuro  it<  the  limit  of  all 
the  steps  that  can  ho  taken  in  the  nianncr  desr-rihed. 

The  following  principle  will  enahle  us  to  find  the  limit  of 
the  sum  of  a  progression  : 

*Z\V,  Principle.  Tf  r  <  1,  the  pow«'r  r"  ran  ho  made 
as  small  as  we  please  by  increasing  the  value  of  7/,  but 
can  never  be  made  equal  to  0. 

Suppose,  for  instance,  that 

_  3  _  1 

^  ~  4  ~  4' 


1 


Then  every   time  we   multiply  by   r  we  diminish  r'*  by 
of  its  former  value ;  that  is, 

r2  =  ^^r={\  -^)r-r-^r, 


8        3   . 

~  4 


►•8 


2  ^     2 

^         4'  ' 


fA  —   ^3   —   /i*o  

'   -  4'   -  ' 


etc. 


etc. 


4'' 

etc. 


Now  let  us  again  take  the  expression  for  the  sum  of  a 

series  of  n  terms,  namely, 

1  _  ^n 

X  =  a- , 

1  —  r 

which  we  may  put  into  the  form 

a  a 


E  = 


1  —  r       1  —  r 


fth 


t 


.hi 


1 


f 


Ml 

i 


.11 


I 


214 


GEOMETRICAL    PROGRESFJOI^. 


If  ?•  is  less  than  unity,  wc  can,  by  the  principle  just  cited, 
make  the  ([uantity  r'*  as  small  as  we  i)leasc  by  inereasing  n 
indefinitely.     Krom  this  it  follows  that  we  can  also  make  the 

term  - — —  r"  as  small  as  we  please. 
1  —  r 

Proof.     Let  ns  pnt,  for  brevity, 


a 


1-r' 
so  that  the  term  under  consideration  is 

If  we  cannot  make  kr'^  as  small  as  we  please,  suppose  s  to 
be  its  smallest  possible  value.     Let  us  divide  s  by  k,  and  put 

No  matter  how  small  s  may  be,  and  how  large  k  may  be, 
v,  or  f,  will  always  be  greater  than  zero.     Hence,  by  the  pre- 

rC 

ceding  principle,  we  can  find  a  value  of  fi  so  great  that  /" 
shall  be  less  than  /.     That  is, 

'■"  <  I- 

Multiplying  both  sides  of  this  inequality  by  ky 

Tern  <  s. 

That  is,  however  small  we  take  s,  we  can  take  n  so  large 
that  kr'^  shall  be  less  than  s,  and  therefore  .s  cannot  be  the 
smallest  value. 

a 


Since 


2  = 


—  ^•r^ 


and  since  we  can  make  hr^  as  small  as  we  please,  it  follows 


that 


Limit  of  2  =  --' — 
1  —  r 


This  is  sometimes  expressed  by  saying  that  when  r  <  1, 


«  +  «r  4-  ar^  +  ar^  +  etc.,  ad  injinitnm. 

his  is  a  convenient 
us  into  error  in  this  case 


a 


1-r' 
and  this  is  a  convenient  form  of  expression,  which  will  not  lead 


GEOMETRICAL   PROGRESSION.  215 

EXERCISES. 

Having  given  the  progression 

1111 

'2  +  4  +  8  +  10  +  '^'" 

of  whicii  the  limit  is  1,  find  how  many  terms  we  must  take  in 
order  that  the  sum  may  differ  from  1  by  less  than  the  follow- 
ing quantities,  namely : 

Firstly,  .001 ;  secondly,  .000  001 ;  thirdly,  .000  000  001. 

To  do  this,  we  must  find  what  power  of  ^^  will  be  less  than  .001, 
what  power  lesa  than  .000  001,  etc. 

What  aie  the  limits  of  the  sums  of  the  following  series : 


I. 


2. 


4- 
6. 


-  +  —  -|-  -g  +  etc.,  ad  iufinitiim. 
o        o         o 

2       4        8  ^  1  ■   .    -s 

-  +  -  -f  - -^  +  etc.,  ad  innnitum, 

-  —  --g  +  as  ~  ^^^">  ^^^  infinitum. 

4       4'^       43 

-  +  — ,  +  — 3  +  etc.,  ad  infinitum. 


9    '   92 
1 


93 


1  1  .... 


a 


a 


h 
1 


Vo  + 


a 


etc.,  ad  infinitum. 


1  {h-  1)2    '    (^,  _  1)3 

2  12         1 

7.  1 1 ^ -„  H ;:  —  etc.,  ad  infinitum. 

m       m^      m^      m'^ 

8.  What  is  that  progression  of  which  the  first  term  is  Vi 
and  the  limit  of  the  sum  8. 


d 


c 


B 


9.  On  the  line  AB  a  man  starts  from  A  and  goes  to  the 
point  c,  half  way  to  B ;  then  he  re- 
turns to  d,  half  way  back  to  A ;  then        , 
turns  again  and  goes  half  way  to  c, 

then  back  half  way  to  d,  and  so  on,  going  at  each  turn  half 
way  to  the  point  from  which  he  last  set  out.  To  what  point 
on  the  line  will  he  continually  approach  ? 


I 

I 


216 


COMPOUND    INTEREST. 


i 


^  I 


i# 


•■'ii 

\ 

4 
I 

-J 

If'  il 


♦  ri;  i 


215.  As  an  interesting  application  of  the  preceding  theory, 
we  may  examine  the  problem  of  finding  the  value  of  a  circu- 
lating decimal.  Such  a  decimal  is  always  equal  to  a  vulgar 
fraction,  which  is  obtained  as  in  the  following  examples : 

I.  AVhat  is  the  value  of  the  decimal 

.373737 ? 

We  find  the  figures  which  form  the  period  to  be  37.  Dividing  the 
decimal  into  periods  of  these  figures,  its  value  is 

_37         3_7     ,   _37 

100  ^  1002  "^  1003  +  etc. 


= ^^  a 


303  +  e*^')- 


100  "^  1C02  "^  1003 
The  quantity  in  the  parenthesis  is  a  geometrical  progression,  in  which 
a  =  :rx^  ,  r  =  ^7^77  •   The  limit  of  its  sum  is  therefore  zr^  •    Therefore  the 


100' 


100 


99 


▼alue  of  the  decimal  is 


37 
99 


This  result  can  be  proved  by  changing  this  vulgar  fraction  to  a 
decimal. 

2.  In  the  case  of  a  decimal  which  has  one  or  more  figures 
before  the  period  commences,  we  cut  these  figures  off,  and 
find  the  value  of  them  and  of  the  circulating  part  separately. 
Thus, 

56363  etc.  =  ~  +  -^  +  ,   "Ln  +  etc. 


10 


1000 
63 


100000 


=  -'-  + 
10  ^  1000 


(^  +  160  +  m^  +  '^') 


63 


"^  To  "^  1000* "99"  ~  10  "^  990 


EXERCISES. 


558 
990 


31 
55* 


To  what  vulgar  fractions  are  the  following  circulating  deci- 
mals equal : 

.111111 ?  2.     .2222 ? 


I. 


3.     .9999 ? 

5.     .454545 ? 

7.     .108108 ? 


4.     .09999 ? 

6.     .2454545 ? 

8.     72454545 ? 


\t 


: 
i 

f 
4i 

COMPOUND    INTEREST. 


217 


ction  to  a 


Coinpouiid  Interest. 

316.  When  one  loans  or  invests  money,  collects  the  inter- 
est at  stated  intervals,  and  again  loans  oi  invesU  this  interest, 
and  so  on,  he  gains  compound  interest. 

Compound  interest  can  always  be  giuned  by  one  who  con- 
stantly invests  all  his  income  derived  from  interest,  i)rovi(kd 
that  he  always  collects  the  interest  wiien  due,  and  is  al)le  to 
loan  or  invest  it  at  tlie  same  rate  as  lie  loaned  his  principal. 

Problem  I.  To  find  the  (unount  of  p  dollars  for  ti 
years,  at  c  per  cent,  conipouiid  interest. 

Solution.  At  the  end  of  one  vear  the  interest  will  be 
i--,  which  added  to  the  principal  will  make  py-  -r-  r^A/' 


If  we  put        p  ==  — -  =:  the  rate  of  annual  gain, 

the  amount  at  the  end  of  the  year  will  be  7?  (1  +  p). 

Now  suppose  this  whole  amount  is  put  out  for  another 
year  at  the  same  rate.  The  interest  will  be  j)  (1  +  p)p,  which 
added  to  the  new  principal  />  (1  ■\-  p)  will  make  />  (1  +  py. 

It  is  evident  that,  in  general,  supposing  the  whole  sum 
kept  at  interest,  the  total  amount  ot  the  investment  will  be 
multiplied  1)y  1  -f  p  each  year.  Hence  the  amount  at  the  ends 
of  successive  years  will  ba 

P  (1  +  P)^    P  (1  +  pY,    ;?  (1  +  pY,    etc. 
At  the  end  of  71  years  the  amount  will  be 

^  (1  +  pY. 

Problem  II.  A  person  puts  out  p  dollars  every  year, 
letting  the  whole  siom  constantly  accurnulate  at  coni- 
pound  interest.  JVJiat  ivill  the  amount  be  at  the  end  of 
n  years? 

Solution.  The  first  investment  will  have  been  out  at 
interest  n  years,  the  second  71  —  1  years,  the  third  71  —  2  years, 
and  so  on  to  the  71^^,  which  will  have  been  out  1  year.  Hence, 
from  the  last  formula,  the  amounts  will  be : 


'  - '    i 


'    t 


I      , 


ti  . 


218 


COMPOUND   INTEREST. 


Amount  of  1st  payment,    ^?  (1  +  p)' 


kl. 

"   2d 

a 

p{\  ^pY-k 

ii 

"   3d 

a 

p{i  +pY-\ 

a 

"   4th 

a 

p{\.  +  pY~\ 

a 

''   5th 

a 

p{i+p)n~i. 

etc. 

etc. 

The  sum  of  the  amounts  is : 

p{l+p)  +p{l-\-py  +  p{l  +  pY4- 


pO^+pY- 


This  is  a  geometrical  progression,  of  whicli  the  first  term  is 
p  (l+p)j  the  common  ratio  1  +  p,  and  the  number  of  terms  )i. 
So  in  the  formula  (4),  §  212,  we  put  p  (1+p)  for  a,  1  +  p  for 
r,  and  thus  tind, 

''^■■'1  +  p  —  1        ^  p 


I    I'tl,    --,3/      ■ 


5,.! 


EXERCISES. 

1.  A  man  insures  his  life  for  $5000  at  the  age  of  30,  pays 
for  his  insurance  a  premium  of  80  dollars  a  year  for  32  years, 
and  dies  at  the  age  of  02,  immediately  before  the  33d  payment 
would  have  been  due.  If  the  company  gains  4  per  cent,  inter- 
est on  all  its  money,  hew  much  does  it  gain  or  lose  by  the 
insurance  ? 

Note.  Computations  of  this  class  can  be  made  with  great  facility  by 
the  aid  of  a  table  of  logarithms. 

2.  What  is  the  present  value  of  a  dollars  due  n  years  hence, 
interest  being  reckoned  at  c  per  cent.  ? 

Note.     If  p  be  the  present  value,  Problem  I  gives  the  equation, 

^  ■*-  loo)  =  ^' 

3.  What  is  the  present  value  of  3  annual  payments,  of  a 
dollars  each,  to  be  made  in  one,  two,  and  three  years,  interest 
being  reckoned  at  5  per  cent.  ? 

4.  What  is  the  present  value  of  n  annual  payments,  of  a 
dollars  each,  the  first  being  due  in  one  year,  if  the  rate  of  in- 
terest is  c  per  cent.  ?  What  would  it  be  if  the  first  payment 
were  due  immediately  ? 


SECOND   PART. 


■►■ 


ADVANCED    COUrvSE. 


*      iiH* 


if] 


i  : 


!    !• 


'Ill 


:!t,^l 


^X:?^ 


BOOK    VIII. 

RELATIONS   BETWEEN  ALGEBRAIC 

QUANTITIES. 


Of  Algobrair  Fiiiictioiis. 

317.  Def.  When  one  quantity  depends  upon  an- 
otlier  in  such  a  way  that  a  change  in  tlie  value  of  the 
one  produces  a  change  in  the  value  of  the  other,  the 
latter  is  called  a  Function  of  the  former. 

This  is  a  more  general  definition  of  the  word  "  function  "  thiin  that 
given  in  ^  49. 

Examples.  The  time  required  to  perform  u  journey  is  a 
function  of  the  distance  because,  other  tilings  being  equal,  it 
varies  with  the  distance. 

The  cost  of  a  package  of  tea  is  a  function  of  its  weight,  Ijc- 
cause  the  greater  the  weight  the  greater  the  cost. 

An  algebraic  expression  containing  any  symbol  is  a  func- 
tion of  that  symbol,  because  by  giving  different  values  to  the 
symbol  we  shall  obtain  different  values  for  the  expression. 

Def.  An  Algebraic  Function  is  one  in  which  tlie 
relations  of  the  quantities  is  expressed  by  means  of  an 
algebraic  equation. 

Example.  If  in  a  journey  we  call  /  the  time,  .9  the  average 
speed,  and  d  the  distance  to  be  travelled,  the  relation  between 
these  quantities  may  be  expressed  by  the  equation, 

(I  =  st. 

Any  otie  of  these  quantities  is  a  function  of  the  other  two, 
defined  by  means  of  this  equation. 

An  algebraic  function  generally  contains  more  than  one 


r' 


M 


I  I, 

I    Ml 


FUNCTIONS. 


r< 


i;  i 


» 


i;-,"ii    I 


i  J 


letter,  and  thoreforo  dejionds  upon  sevonil  quantities.  But  wo 
may  consider  it  a  function  of  any  one  of  tiiese  (juantities,  t^i- 
lected  at  pleasure,  by  .supposing  all  the  other  (piantities  to 
remain  constant  and  only  this  one  to  vary.  For  example,  the 
time  refpiirod  for  a  train  to  run  between  two  points  is  a  func- 
tion not  only  of  their  distance  apart,  but  of  the  speed  of  the 
train.  The  speed  beinp^  sn})posed  constant,  the  time  will  be 
greater  the  greater  the  distance.  The  distance  being  constant, 
the  time  will  be  greater  the  less  the  speed. 

Def.  Th(3  quantities  between  wliich  the  relation  ex- 
pressed hy  a  function  exists  are  called  Variables. 

riiis  term  is  nsfd  because  eucli  quantities  in;      vary  in  value,  as  in 
the  preceding  examples. 

D(f.  An  Independent  Variable  is  one  to  which  we 
may  assign  values  at  jdeasure. 

The  function  is  a  dependent  variable,  the  value  of  which  is 
determined  by  the  value  assigned  to  the  independent  variable. 

Def.  A  Constant  is  a  quantity  which  we  suppose 
not  to  vary. 

RE>r.  This  division  of  quantities  into  constant  and  varia- 
ble is  merely  a  supposed,  not  a  real  one  ;  we  can,  in  an  algebraic 
expression,  suppose  any  quantities  "we  please  to  remain  constant 
and  any  we  please  to  var}^  The  former  are  then,  for  the  time 
l)eing,  constants,  and  the  latter  variables. 

Illustration.    If  we  put 

d,  the  distance  from  New  York  to  Chicago  ; 
,v,  the  average  speed  of  a  train  between  the  two  cities ; 
/,  the  time  required  for  the  train  to  perform  the  jour- 
ney, 

then,  if  a  -nanager  computes  the  different  values  of  the  time  / 
corresponding  to  all  values  of  the  speed  x,  he  regards  d  as  a 
constant,  ,s'  as  an  independent  variable,  and  /  as  a  function  of, v. 
If  he  computes  how  fast  the  train  must  run  to  perform  the 
Journey  in  different  given  times,  he  regards  t  as  the  independ- 
ent variable,  and  s  as  a  function  of  /. 


FUNCTIONS. 


AVlien  we  liavo  any  C(iiuitiun  between  two  viiriuMoi^,  we 
may  ivyard  eitluT  (•!"  tliein  as  an  independent  variable  and  tho 
(itlier  as  a  t'unction. 

KxAMi'LK.     From  tlio  eqnation 

ax  ^-  by  —  c, 

ne  derive  x  =. 


''H 

e 

+ 

) 

a 

a 

ax 

r 

b 

+ 

ir 

y  == 

ill  o'le  of  which  x  is  expressed  as  a  function  of?/,  and  in  the 
other  y  as  a  function  of  .f. 

^18.  Names  are  given  to  particuhir  cLasses  of  functions, 
among  which         following  are  the  most  comm  Jii. 

1.  Def.  A  Linear  Function  of  several  variables  is 
one  in  which  each  term  contains  one  of  the  variables, 
and  one  only,  as  a  simple  factor. 

Example.     The  expression 

Ax  +  By  -\-  Cz 

is  a  linear  function  of  .r,  //,  and  z,  when  A,  B,  and  f'are  quan- 
tities which  do  not  contain  these  variables. 

A  linear  function  diit'ers  from  a  function  of  the  first  degree 
(§  52)  in  having  no  term  not  multiplied  by  one  of  the  varia- 
bles.    For  example,  the  expression 

Ax  +  By  +  C 

is  a  function  of  x  and  y  of  the  first  degree,  but  not  a  linear 
function. 

The  fundamental  property  of  a  linear  iu.iction  is  this: 

//  all  tli6  variables  he  multiplied  hij  a  common  fac- 
tor, the  function  will  he  multiplied  hy  the  same  factor. 

Proof.  Let  Ax  -^r  By  -\-  Cz  be  the  linear  function,  and  r 
the  factor.  Multiplying  each  of  the  variables  x,  y,  and  z  by 
this  factor,  the  function  will  become 

Arx  +  Bry  +  Crz, 
which  is  equal  to         r  {Ax  +  By  +  Cz). 


f 


:  :| : 


1 1 


»'  '  i  • 
I' 


•*    H 


•! 


t 


"f  ii 


I .; 


,       ] 


"■-   ---■  ->.. -..^- 


224 


hVUVTIONH. 


i- 


r, 


Moreover,  r^  liiuuir  function  is  the  only  one  jrJiich  pos- 
sesses this  jn'oprrt!/. 

2.  Drf.  A  Homogeneous  Function  of  several  va- 
j'iablcs  is  one  in  wliieli  each  term  is  of  the  same  degree 
ill  the  variables.     (Compare  §  C)2.) 

ExAMPLi:.  Tlio  expression  ajfi -{•  bx'^ij -{- cifz -\- dz^  is  homo- 
f^eiieous  and  of  tlie  third  degree  in  the  variubles  x,  ?/,  and  z. 

Ukm.  a  Hneur  function  is  a  lioinogeneous  function  of  tlie 
first  degree. 

Fundamental  Property  of  Homogeneous  Functions. 
If  all  the  variables  he  maltlpUcd  bij  a,  eoninion  faetor, 
ami  hjODiogencous  fanetioii  of  the  n*'^  degree  in,  those  va- 
ri((bles  will  be  nuiltiplied  by  the  i<"*  power  of  tJiat  factor. 

rroof.  If  we  take  a  homogeneous  function  and  put  rx  for 
.r,  ry  for  ?/,  rz  for  2;,  etc.,  tlien,  because  each  term  contains  .r, 
y,  or  z,  etc.,  n  times  in  idl  as  a  factor,  it  will  contain  r  n  tiuK-s 
after  the  suljstitution  is  made,  and  so  will  be  multiplied  by  r'K 

3.  D<f.  A  Rational  Fraction  is  tlie  quotient  of  two 
entire  functions  of  the  same  variable. 

A  rational  fraction  is  of  the  form, 

a  -]-  hx  -\-  r.7-2  4-  etc. 


m  -\-  nx  -f  px^  -\-  etc. 

Any  rational  function  of  a  variatle  may  he  expressed  as  a 
rational  fraction.     Compare  §  180. 


Equations  of  the  First  Dej?ree  between  Two 

Variables. 

219.  Since  we  may  assign  to  an  independent  variable  any 
values  we  please,  we  may  suppose  it  to  increase  or  decrease  by 
regular  steps.  The  diiference  between  two  values  is  then 
called  an  increment.     That  is, 

Def.  An  Increment  is  a  quantity  added  to  one 
value  of  a  variable  to  obtain  another  value. 


imniHMHXTS. 


Rem.  If  we  (liininish  the  vuria])lo,  the  increment  k 
nogiiti\e. 

llieorem.  In  ;i  function  of  tlic  first  droTci',  (Hiual  in- 
crenuMits  of  thi?  independent  viii'labh'  cause  etnuil  inciv- 
iiieiits  of  the  function. 

Example.     Let  x  be  an  iiidcpondeut  vuriublo,  and  call  xl 

3 

tliL'  Tunc'tion  ^^x  +  11,  so  tiuit  we  liavo  . 

Xi  =      X  +  11. 

/V 

If  we  give  x  the  successive  vahies  —2,  —1,  0,  1,  2,  ctc.y 
and  find  tlie  e()rre,s[)onding  vahies  of  tlie  runcliou  ii,  they 
\vill  bo 

Values  of  2-,  —2,  —1,  0,  1,  2,  3,  1,  etc. 
"       "  x(.,         8,         0^,     11,     12^,  U,     l-H,  17,     etc. 

We  see  that,  the  increments  of  x  being  all  unity,  those  of 
ij  are  all  1 U 

General  Proof.  Let  an  +  bx  =  c  be  any  equation  of  ihe 
iirst  degree  between  the  variable  x  and  tlu'  function  i(.  St)lving 
this  equation  we  shall  have 

_  c  —  bx  _  c       h 

~       a      ~~  a      a  ' 

Let  us  assign  to  x  the  successive  values, 
r,     r  -\-  h^    r  +  2//,     etc., 

the  increment  being  //  in  each  ease.    The  corresponding  values 
of  the  function  u  will  be 

c       b  c       b         b^        c       b         2b-, 

r,  r n,       r n,     etc., 

a      a         a      a        a  a      a         a 

of  which  each  is  less  than  the  preceding  by  the  same  amount, 

h.     Hence  the  increment  of  u  is  always //,  which  proves 

the  theorem. 

320.   Geometric  Constrxiction  of  a  Relation  of  the  First 
Degree.     The  relation  between  a  variable  x  and  a  function  v, 
of  this  variable  may  be  shown  to  the  eye  in  the  following  way; 
15 


»' 


2i^0 


OmMiaitliJ   (JOi\tiTH  LCI  ION. 


>s 


V  ^ 


•h» 


N 


-J 


N.. 


ts. 


-ir 


^ 


N. 


"r 


I 


i:-: 


^ 


V, 


"v.. 


V. 


Take  a  base  lino  AX,  mark  a  zero  point,  upon  it,  and  from 
tills  zoro  ])()int  lay  off  any  vahu'S  of  x  wo  ploase.  1'iien  at  each 
l)oint  of  tlie  line  ('orrt's})()n(liii<jj  to  a  value  of  .7--  erect  a  vertieal 
line  e(|ual  to  the  correspond! n<,^  value  of  //.  If  n  is  ])ositive,  tin; 
value  is  measured  upward;  if  nen;;itive,  downward.  Tiie  line 
drawn  tiiroui^h  tiie  ends  of  these  values  of  u  will  show,  by  th.- 
distance  of  each  of  its  points  from  the  base  line  AX,  the  values 
of  u  eorres})onding  to  all  values  of  x. 

'Lai  us  take,  as  an  example,  the  ecpiation 

5?^  +  3^  =  10, 


the  solution  of  which  gives     w 

Computing  the  values  of  w  corresponding  to  values  of  x 
from  — 3  to  -f  G,  we  tind  : 

X  =z  -3,     -2,     -1,      0,    +1,      +2,    +3,   +4,   +5,   +G. 
u=  +3|,    +3],       2|,    2,       If,       i       i,  -I,  -1,   -If. 


Laying  off  these  values  in  the  way  just  described,  we  have 
the  above  figure.  Wherever  we  choose  to  erect  a  value  of  //, 
it  will  end  in  the  dotted  line. 

We  note  that  by  the  property  of  functions  of  the  first  de- 
gree just  proved,  each  value  of  w  is  less  (shorter)  than  the  pre- 

3 

ceding  one  by  the  same  amount;  in  the  present  case  by  -•    It 

o 

is  known  from  geometry  that  in  tliis  case  the  dotted  line 
through  the  ends  of  7i  will  be  a  straight  line. 

We  call  this  line  through  the  ends  of  y  the  equation  line. 


OF  EQUATloys   OF   Till:   hlliST   DEdlUCl': 


227 


*i2i,  Wljoii  we  can  onci'  driiw  this  ritnii/^lit  lino,  wo  can 
liiid  t.lio  vuliu'  of  y  t'orrcsjXJiuUn^^  !<>  cvoi'V  value  of  ./•  without 
iisiii;jf  tiio  o({uatioii.  \\  »•  have  only  to  take  tlif  point  in  I  lie 
li;isi'  lino  coiTi'spoinlin^'  to  any  value  of  ;;•,  ami  by  nu.'usurin<5 
ill.'  distance  to  tiie  line,  we  wliall  Iuinc  liie  cniresjiondin';  value 
•if  ti. 

Now  it  is  an  axiom  of  fT^eomelry  that  o\n\  straifi^ht  liin',  and 
only  one,  can  he  drawn  helween  any  two  points.  'I'herefore, 
to  form  any  rehition  of  the  lirst  (le<j^reo  we  please  between  ./; 
and  X,  we  may  take  any  two  values  of  j;  assi«j^!i  to  I  hem  any 
iwo  values  of  u  we  jjlease,  plot  these  two  pair  of  values  of  //  in 
a  (lia<jfram,  draw  tlie  e(|uation  line  (hrou<;li  them,  and  (hen 
measure  olf,  by  this  line,  as  nuiny  more  values  of  //  as  we 
please. 

EXAMPLK.  T.et  it  l)p  required  that  for  ^  =  -f  I  w(^  shall 
have  u  =  -f- 1,  and  for  x  =  -1-5,  u  =  -|-  -i.  What  will  be  the 
values  of?/  eorrespondiiif^  to  x  ■=  —'.],  — '^,  —1,  0,  etc. 

Drawing  the  l)ase  lino  AX  below,  we  layoff  from  1  the  ver- 
tical line  -f  1  in  len^jth,  and  from  the  point  5  the  vertieal  line 


Then  drawinjj:    the   dotted   line  throuj;!!   the  ends,  wi 


measure  olf  dill'erent  values  of  «,  as  follows' 


;r  =  -3, -2,  -1, 


0,  +1,  -f  3,     +3,  +4,     +5,  +{),    etc. 
1,  +li  +2,  +X>i,  +3,  -f-3J,  etc. 


"» 


— H 


EXERCISES. 

1.  Plot  the  equation  2u  -f  ^x  —  (\. 

2.  Plot  a  line  such  that 

for    X  =  —  G    we  shall  liave     u  =  +4, 
for    a:  =r  +  G        '•'         "  u  =  —  4:, 

and  find  the  values  of  w  for  a;  =  1,  2,  3,  4,  and  5. 


I 


228 


GEO  METRIC   CONS  Tli  UCTION 


\  > 


i  ■  X 


%  i 


333.  The  algebraic  problem  corresponding  to  the  con- 
struction of  §  220  is  the  following: 

Having  given  tivo  values  of  y  corresponding  to  two 
given  values  of  X,  it  is  required  to  construct  an  equation 
of  the  first  degree  such  that  these  two  pairs  of  values 
sliall  satisfy  it. 

Example  of  Solution.  Let  the  requirement  be  that  of  the 
equation  plotted  in  the  preceding  example,  namely, 

for    a;  —  1     we  must  have    w  =  1, 
for    2:  =:  5        "         "  u  =  3. 

The  problem  then  is  to  find  such  values  of  a,  b,  and  c,  that 
in  the  equation 

ax  +  bu  =  c,  (1) 

we  sh.'ill  have  u  =  1  for  x  =  1,  and  u  =  3  for  x  =  5.     Sub- 
stituting these  two  pairs  of  values,  "ve  find  that  we  must  have 

axl  -^  bxl  =  c, 
axT)  +  Z»x3  =  c; 

or  a  -\-  b  —  c, 

5a  +  3b  =  c. 

Here  a,  b,  and  c  are  the  unknown  quantity  s  whose  values 
are  to  be  found,  and  as  we  have  only  two  equations,  we  cannot 
find  tliem  all.     Let  us  therefore  find  a  and  b  in  terms  of  c. 

Multiplying  tlie  first  equation  by  3,  and  subtracting  the 
j)roduct  from  the  second,  we  have 

9/ 


la  =i 


2c 


or    a  =  —  c. 


Multiplying  the  first  equation  by  5,  and  subtracting  the 
second  from  the  product,  we  have 

2b  =  4:0    or     b  =  2c. 

Substituting  these  values  of  a  and  b  in  (1),  we  find  the  re- 
qivred  equation  to  be 

2c?*  —  ex  =z  c. 

We  may  divide  all  the  terms  of  this  equation  by  c  (§  120, 
Ax.  Ill),  giving 

2u~x  =  1, 


OF  EQUATIONS   OF   THE  FIRST  DEO  REE. 


220 


thus  showing  that  there  is  no  need  of  using  c.     The  solution 
of  this  equation  gives 

1  +  X 


u 


'Z 


from  which,  for  x  ■=  — 3,  — 2,  — 1,  etc.,  we  shall  find  the  same 
values  of  u  which  we  found  from  the  diagram. 

EXERCISES. 

Write  equations  between  x  and  y  which  shall  be  satisfied 
by  the  following  pairs  of  values  of  x  and  y. 

1 .  For  X  =1  'H,  y  =:  1',  and  for  x  =  5,  y  =  —  1. 

2.  For  X  =:  —  2,  y  =  —  1  ;  and  for  x  =z  -{-2,  y  =  +1. 

3.  For  X  =  —  5,  y  =z  -f-2;  and  for  x  =    i-5,  y  =:  — 2. 

4.  For  a;  =:  0,  y  =  —  7  ;  and  for  x  =  15,  y  =  ^. 

5.  For  X  =  26,  y  =  2  ;  and  for  x  =z  30,  y  =1  3. 

333.  Geometric  Solution  of  Two  Equation?,  ivitli  Two  Un- 
known QnantitieH.  The  solution  of  two  equations  with  two 
unknown  quantities  consists  in  finding  that  one  pair  of  values 
Avhich  will  satisfy  both  equations.  If  Ave  lay  off  on  the  base 
line  the  required  value  of  x,  the  two  values  of  //  corresponding 
to  this  value  of  a;  in  the  two  e(|uations  must  be  the  same  ;  that 
is,  the  two  cf/uatfon  lines  must  cross  each  other  at  the 
point  thus  found.     Hence  the  following  geometric  solution: 

I.  Plot  the  two  cqu<ttions  from  the  same  base  line  and 
zero  point. 

II.  Continue  the  equation  lines,  if  necessary,  until 
they  intersect. 

III.  Tlie  distance  of  the  point  of  intcj'section  from  the 
h((se  line  is  tlie  value  of  y  irhich  satisfies  hoth  equations. 

IV.  TJie  distance  of  the  foot  of  the  y  line  from  the 
zero  point  is  the  required  value  of  x. 

EXERCISES. 

Solve  the  following  e([uations  by  geometric  construction : 

1.  X  —  2u  =  3,     2x  4-  u  =  5. 

2.  2u  ■}-  Hx  =  4,    3u  +  X  =  1, 


■i' 


►'■ 


,  ! 


r 


I    1 


'  il 


i'  ■ 


"! 


230 


NOTATION   OF   FUNCTIONS. 


\  M 


3;'>, 


224.  Geometric  Explanation  of  Equivalent  and  Incon.nfit- 

e)/t  Equations.     If  we  have  two  equivalent  equations  (§  200), 

each  value  of  x  will  give  the  same  value  of  the  other  quantity 

u  or  I/.     Hence  the  two  lines  representing  the  erjuation  will 

coincide  and  no  definite  point  of  intersection  can  be  fixed. 

If  the  two  equations 

au  -\-  bx  r=  c, 

a'u  -\-  b'x  =  c', 

are  inconsistent  we  shall  have  (§  142), 

b  _  b^^ 

a  ~  a' 

If  ?i  be  any  increment  of  .r,  the  increments  of  u  in  the  two 

equations    (§219)  will   be A  and ,h   Therefore   these 

^  a  a 

increments  will  be  equal,  and  the  two  equation  lines  will  be 
parallel.     Hence, 

To  iiiconsisteiht  equations  correspmid  parallel  lines, 
which  have  no  point  of  intersection. 

If  the  two  equations  are  equivalent  (§  141,  143),  their  lines 
will  coincide. 

Notation  of  Functions. 

225.  In  Algebra  we  use  symbols  to  express  any  numbers 
whatever.  In  the  higher  Algebra,  this  system  is  extended 
thus : 

We  may  use  a,ny  symbol,  having  a  letter  attached  to 
it,  to  express  a  function  of  the  quantity  represented  hy 
that  letter. 

Example.  If  we  have  an  algebraic  expression  containing 
a  quantity  x,  which  we  consider  as  a  function  of  x,  but  do  not 
wish  to  write  in  full,  we  may  call  it 

F{x),    or    0(;r),     or     [.r],     or    Ax, 

or,  in  fine,  any  expression  we  please  which  shall  contain  tlie 

symbol  X,  and  shall  not  be  mistaken  for  any  otiier  expression. 

In  tlie  first  two  of  the  above  expressions,  the  letter  x  is  enclosed  in 
l)arenthcseH,  in  order  that  the  expression  may  not  be  mistaken  for  x  mul- 
tiplied by  F,  or  (p.  The  parentheses  may  be  omitted  when  the  reader 
knows  that  multiplication  is  not  meant. 


NOTATION    W  FUNCTIONS. 


231 


The  fundamental  principle  of  the  functional  notation  is 
this : 

Wlien  a  symbol  with  a  letter  attached  represents  a 
fuiictlorhtheri,if  we  substitute  any  oilier  (/uanfity  for 
till'  letter  attached,  the  combination  will  represent  the 
function  found  by  substituting  that  other  quantity. 

Example.  Let  us  consider  the  expression  ax^  +  I  as  a 
function  of  x,  and  let  us  call  it  0  {x),  so  that 

^  (x)  —  ax^  +  b. 

Then,  to  form  0  {ij),  we  write  y  in  place  of  x,  obtaining 

<A  {y)  =  f^  +  ^• 
To  form  (p  {x  +  y),  we  write  x-\-ym  place  of  x,  obtaining 

0  (a;  4-  ?/)  =  a  {x  +  yf  +  b. 

To  form  (p  {a),  we  write  a  instead  of  x,  obtaining 

(})  (a)  =  cfi  4-  b. 

To  form  </>  («/),  we  put  ay^  in  place  of  x,  obtaining 

0(«2/3)  =  a{ay^f  +  b  =  ay  +  b. 

The  equation  (p  {z)  =  0  will  mean 

az^  ^  b  =  0. 


;  ■  1!   f ! 


i   '  1 


EXERCISES. 

Suppose  (p  {x)  =  ax^  —  a%  and  thence  form  the  values  of 

I.     cp{y).  2.     (p{z).                     3'     "Pi^y)- 

4.     <p{x-\-y).  5.     (p{x-\-a).            6.     (/jCt-^). 

7.     rp{x-{-  ay).  8.     </)  (a:  -  rti/)-           9-     "P  i^")- 
Suppose  i^(.i')  =  xa^,  and  thence  form  the  values  of 

10.     F(ij).  II.     F{2y).               12.     /M'3//). 

13.     F{'>-^y)-  ^4.     F{x-y).         15-     ^(1)- 

Suppose  /'  (:c)  =  a;2,  and  thence  form  the  values  of 

16.    /(I).  17.    /O-^).  ^8-    /(•'■')• 

19.    /(.2;4).  20.    f{jJ'). 


21.    /(2;'0. 


-   i 


J 

■  i'! 


^;H  • 


r'l 


232 


FUNCTIONS   OF  SEVERAL    VARIABLES. 


4 


v^» 

^*«i 


?-r 


2  2.  Prove  that  if  avo  put  (t>{x)  ■=.  «*,  we  shall  have 

Let  us  put  0  {m)  =  m  (m  —  1)  {7n  —  2)  (???  —  3) ;   tlieuce 
form  tlie  values  of 

23.     0(G).  24.     0(5).  25.     (?)(4). 

26.     0(3).  27.     0(2). 

29.     0(0).  •  30.     0(— 1). 


28.        0(1). 

31.     0(-2). 


Functions  of  Several  Variables. 

22Cu  An  algebraic  expression  containing  several 
quantities  may  be  represented  by  any  symb(^l  having 
the  letters  wliicli  represent  the  quantities  attached. 

Examples.    We  may  put 

0  {x,  y)  =  ax  —  hj, 
the  comma  being   inserted  between   x  and  y,  so  that  their 
product  shall  not  be  understood.     We  shall  then  have, 

0  (???,  n)  =  am  —  hn. 
0  {y,  ^)  =  ay  —  bx, 
the  letters  being  simply  interchanged. 

(p(x  +  y,  x-y)  =  a  {x  +  ?/)  -h{x-  y) 

=  (a—  b)  X  +  («  +  §) y, 
0  (a,  b)  r=r  a^  -  b\ 
0  [b,  a)  =  ab  —  ba  —  0. 
^  [a  -\-  b,  ab)  =  a  {a  +  b)  —  ab^. 
0  (a,  a)  =  a^  —  ba. 
etc.  etc. 

If  we  put  0  {a,  b,  r)  =  2«  +  36  —  5c,  we  shall  have 
0  (.r,  z,  y)  =  2x  +  3^  —  5y. 
0  {z,  y,  x)  =  2z  +  3y  —  6x. 
(f)  (m,  7/1,  —  m)  =  2m  -{-  3m  +  5m  =  10m, 
0(3,8,  6)  =  2-3  +3.8-5.6  =  0. 


EXERCISES, 


Let  us  put 


0(-^',  y) 

f(^>  y) 

f{^,y^z) 


3x  —  4y, 
ax  4-  by, 
ax  +  by  ■ 


abz. 


USE   OF  INDICES. 

Thence  form  the  expressions : 

I.     (f){y,x).  2.     (I){a,b). 

4.     0(^,3).  5.     0(10,1). 

7.    ./(^«)-  8.    f{y,x). 

10.    f{q,  -p)'        II-    /(2!,  ^,y)' 

13.    f{a,b,c).         14.    f{a\bKc'^). 

15.  /(— «,   —  ^   —  «J). 

w(m  — l)(m- 
Let  us  put     (m,  ?i)  =  — 7 t-^t — 

Find  the  values  of 

16.  (3,3).  17.  (4,3). 
19.  (0,  3).  20.  (7,  3). 
22.     (2,  -1).  23.  (3,  -2). 

Use  of  Indices. 


233 


3-     0  (3,  4). 
6.    y(a,  ^>). 

12.    /(^  a,  2). 


l2) 

18.  (5,  3). 

21.  (8,3). 

24.  (4,  -3). 


S^Oa.  Any  number  of  different  quantities  may  be 
represented  by  a  common  symbol,  the  distinction  being 
made  by  attaching  numbers  or  accents  to  the  symbol. 

EXAMPLES. 

1.  Any  n  different  quantities  may  be  represented  by  the 
symbols,  Px,  p^,  p^,  -  >  '  >  Pn- 

2.  A  producer  desires  to  have  an  algebraic  symbol  for  tho 
amount  of  money  which  he  earns  on  each  day  of  the  year.  If 
he  calls  q  what  he  earns  in  a  day  he  may  put : 

q^     for  the  amount  earned  on  January    1, 
q^        ''  "  "  '"         2, 

etc.       "  "  "  "     etc., 

q.,       -  "  "  "       31, 

q.^„       <*  "  "      February  1; 

and  so  on  to  the  end  of  the  year,  when 

^3  5  5  will  be  the  amount  for  December  31. 

Def.  The  distinguishing  numbers  1,  2,  3,  etc.,  are 
here  called  Indices. 

A  symbol  with  an  index  attached  may  represent  a 
function  of  the  index,  as  in  the  functional  notation. 


i 


r 


^:i 


Ml    I 


234 


USE   OF  INDICES. 


% 


f  f 


H 


I. 


EXERCISES. 

Let  us  put  at  =  t{t  +  1).    Then  find  the  vahie  of 

1.  (Iq  +  r/i  +  f^2  + +  «io- 

2.  Prove  the  following  C(in;ition8  by  computing  both  mem- 


bers: 


^1  +  ^^2   =  .>  ^^8- 


4 

;3 


G 

If  we  put  iS'i  =:  1  -f  2  +  3  .  .  .  .  +  i",  we  shall  have 

8^  ^  1. 

iS-g  =  1  +  3  =  3. 

^'3  =  1  +  2  +  3=:=  6,  etc.,  etc. 

Using  the  preceding  notation,  lind  the  values  of  the  ex- 
pressions : 

5.     2>V5  -  «g.  6.     S^Vg  -  r/g. 

337.  Sometimes  the  relations  between  quantities  distin- 
guished by  indices  are  represented  by  equations  of  the  tirst 
degree.    The  following  are  examples: 
Let  us  have  a  series  of  quantities, 

yio,    ylj,    J  2,    J  3,    A^,    etc., 
conn(?ctcd  by  the  general  relation, 

Aix  —  Ai  +  Ait.  {a) 

'      It  is  required  to  express  them  in  terms  of  Aq  and  A^. 

We   put,  in   succession,  /  =  1,  i  =z  2,  i  =  3,   etc.     Then, 
when  i  =  1,  we  have  from  (a), 

Aq  =  A^  -\-  Aq. 

AYhen   i  =  2,        A.^  =  A.,  -{-  A^  =  2.1,  +    A^. 

i  =  3,        A^  =  aI  +  A.,  =  3yl,  +  2Ao. 

i  =  4,         A^=^  A^  +  ^13  =  5J,  +  3^0. 

/  =  5,         J«  r.  Jg  +  .1^  =  8.1,  4-  5ylo, 

and  so  on  indefinitely. 


MISCELLANEOUS  FUNCTIONS. 


235 


EXERCISES. 
I.    If  Ai^\   =   Ai  —  -ii-1, 

wliiit  will  be  the  values  ot  A^    . . .  Aw,  and  in  what  way  may 
all  subsequent  values  be  determined? 

2.  If  Ai^t  =  2Ai  —  /io, 

riiiu  A  2  to  J  5  in  tenns  of  .i^  and  ^Ij. 

3-  If  An  =  iAi  +  Ai_i,    find  A^  to  ^g. 

4.  If  Ai  =  Ji_i  4-  h, 

lind  the  sum  J „  +  J ^  -f  .Ig  -f  .  .  .  .  -(-  J^,  in  terms  of  J,,, 
//  and  n.     (Comp.  §  209,  Prob.  V.) 

5-  If  Ait  =  rAi, 

find  ^1  +  Jg  -f-  /I3  4-  .  .  .  .  +  yl„,  in  terms  of  Aq  and  r. 

6.   If  yJifi  =  ilvli  +  ylj-i, 

find  A2,  A^,  .  .  .  .  Aq,  in  terms  of  ^^  and  A^. 


I    Mi     !'  I 


Miscellaneouti  Fuiictions  of  Numbers. 

338.  We  present,  as  interesting  exercises,  certain  elemen- 
tary forms  of  algebraic  notation  much  used  in  Mathematics, 
and  which  will  be  employed  in  the  present  work. 

1.  When  we  have  a  series  of  symbols  the  number 
of  which  is  either  indeterminate  or  too  great  to  be  all 
written  out,  w^e  may  write  only  the  first  two  or  three 
and  the  last,  the  omitted  ones  being  represented  by  a 
row  of  dots. 

Examples.  a,  h,  c,  .  .  .  .  t, 

Xj      ^9      O9      •      •      •      •      /vO^ 

J  •        /C*        •       •       •       •       ftm 

n  being  in  the  last  case  any  number  gi-eater  than  2. 
The  number  of  omitted  symbols  is  entirely  arbitrary. 

EXERCISES. 

How  many  omitted  expressions  are  represented  by  the  dots 
in  the  following  series: 


I.         i.,    /v,    O,   •    •    •    •    it* 


2.  X,     /4,     Of     .     .     •     •     ?i    ■—     /V» 


r  i 
i  - 
i 


! 

J   ! 


i!  il 


23G 


MltiCELLANEO  UlS   FUNCTIONS. 


4.  w,  )i  —  \y  n  —  11,  ....  n  —  5. 

5.  ;/,  )i  —  Ij  u  —  'Z,  .  .  .  .  n  —  .s  —  1. 

6.  n,  n  —  l,  n  —  2,  .  .  .  .  n  —  s  +  1. 

What  will  be  (hu  last  term  in  the  series: 

7.  2,  3,  4,  etc.,  to  n  terms. 

8.  n,  ?i  —  1,  71  —  2,  etc.,  to  s  terms. 

9.  2,  4,  G,  etc.,  to  k  terms. 

2.  Product  of  the  First  n  Numbers.    The  symboi 

is  used  to  express  the  product  of  the  first  n  numbers. 

l'2-3  .  .  .  .  n. 

Thus,  1 !  :=  1. 

2!  =  1-2  =  2. 
3!  =  1.2-3  =  G. 
4!  =  1.2.3.4  =  24 
etc.  etc. 

It  will  be  seen  that   2!  =  2-1! 

3!  =  3.2! 
And,  in  general,        n\  =:z  n  {71  —  1) ! 
whatever  number  n  may  represent. 


EXERCISES 

Compute 

the  \ 

alues  of 

I. 

5! 

2. 

6! 

4- 

7! 
3!  4! 

5- 

8! 
3!  5! 

6. 

Prove 

the 

equation 

2.4.G.8.. 

7. 

Prove 

thai 

,,  when  n  is  even. 

81 


2n  =z  2»?il 


n^  _  w  (^  —  2)  (?i  —  4) .  .  .  .  4-2 

3.  Binomial  Coefficients.    The  binomial  coefficient 

n{n  —  \){n  —  2)  .  .  .  .  to  s  terms 

r2.3. .  .  .  s 

is  expressed  in  the  abbreviated  forai, 


MISCELLA NEO  US  FUNCTIONS. 


237 


(:)• 


the  parentheses  being  used  to  show  that  what  is  meant 


71 

is  rot  the  fraction  -• 

s 


EXAMPLES, 
:   3. 


(?)  =  I 

(D 


"  i. 2.3.4.5 

n 


=  21. 


(i) 


n  {n  —  1)  {71  —  2) 
1.2.3 

n(7i  —  1)  ....  2.1  _  ^ 
i. 2.3  ....  w       ~ 

{71  +  4)  {}i  +  3)  {71  -t-  2) 
1.2.3 


EXERCISES. 


Compute  the  vahies  of  the  expressions : 

••  ©  -^  (!)  -  it)  -  (D  H-  (!)  -^  (!)  -  (!)  +  Q' 
'■  (I)  ^  (!)  ^  (I)  -  (!)  -  (!)■ 


Prove  the  formulae : 

^'     V2/~2!3! 

/^^  +  1\  _  ^M^  1  /w\ 
5-     Vs^l)  —  7+1  \5/' 

^  (")+(;:)-m 
«•  (3) + c) = m- 


s)  ~  s\  in  —  .s)! 


(f)  -  (1)  =  ("4-)' 


i , 


,4- 


ri 


i  ;  ■ 


■  I'f 


!  ,    ( 


'«• 


BOOK    IX. 
rifE     THEORY     OE    NUMBERS. 


I,       ' 


CHAPTER    I. 

THE     DIVISIBILITY     OF     NUMBERS. 

?*il).  Def.  The  Theory  of  Numbers  is  a  branch 
of  nuitlR'inatics  wliicli  treats  of  the  properties  of  integers. 

Ihf.  An  Integer  is  any  whole  number,  ijositivc  or 
negative. 

In  the  tlicory  of  numbers  the  word  number  is  used  to  ex- 
press an  integer. 

Def.  A  Prime  Number  is  one  which  has  no  divi- 
sor except  itself  and  unity. 

The  series  of  prime  numbers  are 

2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  etc. 

Def.  A  Composite  Number  is  one  which  may  be 
expressed  as  a  product  of  two  or  more  factors,  all 
greater  than  unity. 

Rem.  Every  number  greater  than  1  must  be  either  prime 
or  composite. 

Bef.  Two  numbers  are  prime  to  each  other  when 
they  have  no  common  divisor  greater  than  unity. 

Example.  The  numbers  24  and  35  are  prime  to  each 
other,  though  neither  of  them  is  a  prime  number. 

Rem.  a  vulgar  fraction  is  reduced  to  its  lowest  terms  when 
numerator  and  denominator  are  prime  to  each  other. 


DIVISIBILITY  OF  NUMliF.IlS. 


DIviHioii  into  Prime*  Factorrt. 


239 


^30.  Every  composite  niniil»i'r  may  hy  tlclinition  1)0  di- 
vided into  two  or  more  factors.  If  any  of  these  factors  are 
c'oinposito,  they  may  be  again  divided  into  oilier  factors. 
When  none  of  the  factors  can  be  further  divided,  tlicy  will  ail 
lie  prime.     Hence, 

Tni;oREM.  Every  composite  nimthc?'  may  be  divided 
iitfo  prime  factors. 


Example. 


Whence, 


180  =  0-20, 

9  =  ;^3, 

!^0  =  4-5  =  3.3-5. 
180  =  2-3-;3-3-5  =  22.32.5. 


Cor.  1.  Because  eve.ry  numljer  not  prime  is  composile, 
;ind  because  every  composite  number  may  be  divided  into 
prime  factors,  we  conclude:  Every  number  is  cither  prime 
iir  divisible  by  a  prime. 

Cor.  2.  Every  number,  prime  or  composite,  may  be  ex- 
pressed in  the  form 

2)'^(fn  etc.,  {a) 

where  p,  q,  r,  etc.,  are  different  prime  numbers ; 

«,  /3,  y,  etc.,  the  exponents,  are  positive  integers. 

Rem.  If  the  numl)er  is  prime  there  will  be  but  one  factor, 
namely,  the  number  itself,  and  the  exponent  will  be  unity. 

EXERCISES. 

Divide  the  following  numbers  or  products  into  their  prime 
factors,  if  any,  and  thus  express  the  numbers  in  the  form  {a)  : 

I.     24.         2.     72.         3.     200.         4.     109.         5.     22o. 
6.     250.       7.     91.         8.     U3.         9.     300.        10.     217. 
II.     3072.  12.     1.2-3-4-5.0.7-8.9. 

Rem.  In  seeking  for  the  prime  factors  of  a  number,  it  is 
never  necessary  to  try  divisors  greater  than  its  square  root,  for 
if  a  number  is  divisible  into  two  factors,  one  of  these  factors 
will  necessarily  not  exceed  such  root. 


>  1' 


t  i  f 


^  F      '    li 


'    '  '      . 

i       '    'il 


■   I 


f 


.,111 


240 


DlVlsnnUTY   OF  NUMUEUS. 


•*i 


Coiniiioii  Divisors  of*  Two  Niiiiilx'rs. 

*y»\\,  TiiKORKM  I.  //'  tivo  nitinbers  liarc  <l  common 
factor,  their  siuii  will  hauc  that  same  /'actor. 

rruuf.    Let     a  ha  tho  (.'Oiunion  I'jictor  ; 

?/i,  the  })r()diict  of  nil  tho  other  factors  in  tho 

one  nuiiiher; 
Hi  tlie  correspoudiiig  product  in   the  otlicr 

niiinbcr. 

Then  the  two  niunbers  will  bo 

am    and    an. 

Their  sum  will  be  a  {tii  +  n). 

Because  m  and  n  are  whole  num))er.s  m-^n  will  also  be  u 
whole  number.     Therefore  a  will  lie  a  factor  of  am -{-an. 

Theorem  II.  If  two  numhcrs  have  a  codidioh  factor, 
their  difference  will  have  the  same  factor. 

Proof.    Almost  the  same  as  in  the  last  theorem. 

Cor.  If  a  number  is  divisible  by  a  factor,  all  multi})les  will 
be  divisible  by  that  factor. 

Rem.    The  preceding  theorems  may  be  expressed  as  foil  ;ws : 

If  two  numbers  are  divisible  by  the  same  divisor, 
their  sum,  difference,  and  multiples  are  all  divisible  bij 
that  divisor. 

Rem.  If  one  number  is  not  exactly  divisible  by  another,  u 
remainder  less  than  the  divisor  will  be  left  over.     If  we  put 

Z>,  the  dividend; 
d,  the  divisor; 
q,  the  quotient ; 
r,  the  remainder; 

I)  =  dq  -\-  r, 
D  —  dq  =  r. 

Example.    7  goes  into  GO  9  times  and  3  over.    Hence 

this  means 

G6  =  7-9  +  3,     or    66-7-9  =  3. 


we  shall  have, 
or 


DfVJsmrrjTY  of  numukhs. 


241 


joniinufh 


2li*i,  Pkomlkm.  To  find  Ike  greatest  lunninnii  divisor 
of  tiro  numhcrs. 

Lxit  m  uiid  n  be  the  niiinhers,  and  let  m  bo  the  ^rciilcr. 

1.  Divide  ;//  by  n.  If  tlie  reinuinder  is  /ato,  n  will  !)«•  llic 
divisor  re(|iiired,  because  every  nund)er  divides  itself.  11'  tlicii' 
IS  a  reinuinder,  let  q  be  the  (juotient  and  r  the  remainder. 

Then  m  —  nq  =  r. 

Let  (I  be  the  common  divisor  r('((uired. 
Heeause  in  and   //   are  both   divisible  by  rf,  m  —  uq  must 
also  be  divisible  by  d  (Theorem  II).     "therefore, 

r  is  divisible  l)y  d. 

TIcnce  every  common  divisor  of  m  and  n  is  also  a  common 
divisor  of  ;i  and  r.     Conversely,  because 

m  =  nq  -\-  r, 

every  common  divisor  of  n  and  r  is  also  a  divisor  of  m.  There- 
fore, the  greatest  common  divisor  of  m  and  ii  is  the  same  as 
the  greatest  common  divisor  of  n  and  r,  and  we  proceed  with 
these  last  two  numbers  as  we  did  with  ni  and  n. 

2.  Let  r  go  into  7i  q'  times  with  the  remainder  r'. 
Then  7i  —.  vq  +  r', 

or  n  —  rq'  =  r'. 

Then  it  can  be  shown  as  before  that  d  is  a  divisor  of  r',  and 
therefore  the  greatest  common  divisor  of  r  arid  r'. 

3.  Dividing  r  by  ?•',  and  continuing  the  process,  one  of  two 
results  must  follow.     Either, 

«.  We  at  length  reach  a  remainder  1,  in  which  case  the 
two  numbers  are  prime  ;  or, 

(i.  We  have  a  remainder  which  exactly  divides  the  pre- 
ceding divisor,  in  which  case  this  remainder  is  the  divisor 
required. 

To  clearly  exhibit  the  process,  we  express  the  numbers  m, 
n,  and  the  successive  remainders  in  the  following  form  : 
16 


M    ! 


K»' . 


I    ; 


242 


GREATEST   COMMON  DIVISOR. 


HI* 


m  =  n-q  +  r, 
u  =  r-q   +  r', 
r  =z  r'-q"  +  r", 
r'  =  r"'q"'  -\r"', 
etc.  etc. 


{r  <  n) ; 

(r  <  r) ; 

(/■"<r'); 

(/•'"  <  r") ; 

etc., 


until  wo  reach  a  remainder  equal  to  1  or  0,  when  the  series 
terminates. 

EXERCISES. 

I.  Find  the  G.  C.  D.*  of  ^40  and  155. 

Dividend.       Div.  Quo.    Rom. 

240  =  155-1  +  85. 
155  =    85-1  +  TO. 

85  —    70.1  +  15. 

70  -    15-4 -f  10. 

15  =     10-1  +    5. 

10  =      5-2. 

Therefore  5  is  the  greatest  common  divisor. 

Note.  Let  the  student  arrange  all  the  following  exercises  in  the 
above  form,  first  dividing  in  the  usual  way,  if  he  tinds  it  necessary. 

Find  the  greatest  common  divisor  of 

2.  399  and  427.  3.  91  and  131. 

4.  8  and  13.  5.  1000  and  212. 

6.  799  and  1232.  7.  800  and  1729. 

8.  250  and  025.  9.  1000  and  370. 

10,  If  p  be  a  number  less  than  n  and  prime  to  /?,  show  that 
11  —  p  is  also  prime  to  n. 

II.  If  p  be  any  number  less  than  u.  the  greatest  common 
divisor  between  n  and  p  is  the  same  as  that  between  n  and 
n  —])• 

12.  If  n  is  any  odd  number,  - —  and  —  ;r —  are  both 
prime  to  it. 

Corollarieff.  1.  When  two  numbers  are  divided  bv  their 
greatest  common  divisor,  their  quotients  will  be  prime  to  each 
other. 


*  The  letierrf  O.  C.  D.  are  an  abbreviation  for  Greatest  C-ommon  Divisor 


OEARINQ    OF    WIIKKLS. 


t^4;j 


ion  Divisor. 


2.  Conversely,  if  two  numbers,  )i  and  //,  prime  to  each 
otlier,  are  eaeli  multiplied  by  any  number  (/,  then  d  will  be  tlie 
(i.C.D.  of  dn  and  dn. 

!^iJo.  (rearing  of  WhceU.  An  interesting  problem  con- 
nected with  the  greatest  com- 
mon divisor  is  afforded  by  a 
Ciijumon  pair  of  gear  wheels. 
Ia'I  there  be  two  wheels,  the 
(tiiu  having  m  teeth  and  the 
otlier  n  teeth,  gearing  into  each 
otlier.  If  we  start  the  wheels 
with  a  certain  tooth  <»f  the  one 
a,2;iinst  a  certain  tootli  of  the 
other,  then  we  have  the  questions: 

(I.)  TIow  many  revolutions  must  each  wheel  make  before 
the  same  teeth  will  again  come  together  ? 

(■i.)  With  how  many  teeth  of  the  one  will  each  tooth  of  the 
otlier  have  geared? 

Let  q  be  the  required  num])er  of  turns  of  the  lirst  wheel, 
having  m  teeth. 

Let^j  he  the  required  number  of  turns  of  the  second,  hav- 
ing n  teeth. 

Then,  because  the  first  wheel  has  m  teeth,  qm  teeth  will 
have  geared  into  the  other  wheel  during  the  q  turns.  In  the 
same  way,  pn  teeth  of  the  second  wheel  will  have  geared  into 
the  first.  But  these  numbers  must  be  ecpial.  Tlierefore, 
when  the  two  teeth  again  meet, 

^M  —  qm. 

Conversely,  for  every  pair  of  numbers  of  revolutions  p  and 
'/.  which  fulfil  ire  conditions, 

pti  =  qm, 

the  same  teeth  will  come  together,  because  eacli  wheel  will 

liiive  made  an  entire  number  of  revolutions.     This  equation 

gives 

p  _  m 

q        n 


}'i^ 


'III 
i  ' 

H 

i 

:     J 

'i 
t 


f 


244 


OEAlilNG    OF    WHEELS. 


h 


W 


7)1 

Tlcnce,  if  we  reduce  the  fraction  —  to  its  lowest  terms,  we 

71  ' 

sliall  liave  the  smallest  number  of  revolutions  of  the  respective 
v/hcels  which  will  bring  the  teeth  together  again. 

To  answer  the  second  question  : 

After  the  first  wheel  has  ma(r>  q  revolutions,  qm  of  its  teeth 
have  passed  a  fixed  point.  Any  one  tooth  of  tlie  other  wheel 
gears  into  every  7i^'^  passing  tooth  of  the  first  wheel.    Thcrefuiv 

any  such  tooth  has  geared  into  —  teeth  of  the  first  wheel, 

that  is,  into  j)  teeth,  because,  from  tlie  last  equation, 


or 


qm 
n 

=  V' 

If  d  be  the  G. 

C. 

D. 

of  m 

7)1 
71 

p 

and  71,  then 

—  dp, 

—  dq  ; 
m 

n 

Therefore  each  tooth  of  the  one  wheel  has  geared  into  only 
every  d^^'-  tooth  of  the  other. 

In  the  figure  on  the  preceding  page,  7^^=  21  and  n  =  (i. 
Hence,  d  =  3,  and  each  tooth  of  the  one  will  gear  into  eveiv 
third  tooth  of  the  other.  The  numbers  on  the  large  wheel 
show  the  order  in  which  the  gearing  occurs. 

How  long  soever  the  wheels  run,  the  same  contacts  will 
be  repeated  in  regular  order.  Hence,  //  each  tooth  of  ilw 
one  ivJieel  must  gear  ivitJi  every  tooth  of  the  other,  ilir 
nuDihers  7)i  and,  7i  must  he  prime  to  each  other. 

E  X  F,  R  c  1  s  E  s . 

I.  If  one  wheel  has  40  teeth  and  the  other  10,  show  how 
they  will  run  together. 

Sliow  tlie  same  thing  for  the  following  cases: 


2.      7)1  =    < 


7'>, 


71 


15. 


4.     m  =  36,  71  =  25. 


3.     7)1  =  24,  71  =  18. 
5.     7)1  =  24,  71  =  7. 


NUMBERS  AND    Til  FAR   DIGITS. 


245 


Relations  of  Numbers  to  their  Digits. 

234.  Ill  our  ordinary  method,  of  expressing  numbers,  the 
second  digit  toward  tlie  right  expresses  lO's,  the  third  lOU's, 
etc.  That  is,  each  digit  expresses  a  power  of  10  correspond- 
ing to  its  position. 

Def.  Tlie  number  10  is  the  Base  of  our  scale  of 
iiunieration. 

Note.  The  base  10  is  entirely  arbitrary,  and  is  supposed 
to  have  originated  from  the  number  of  the  thumbs  and  fingers, 
these  being  used  by  primitive  people  in  counting. 

Any  other  number  might  ec^ually  well  have  been  chosen  as 
,a  base,  but  in  any  case  we  sliould  need  a  number  of  separate 
cliaracters  (digits)  equal  to  the  base,  and  no  more. 

Had  8  been  the  base,  we  should  have  needed  only  the 
digits  0,  1,  2,  etc.,  to  7,  and  ditl'erent  combinations  of  the 
digits  would  have  represented  numbers  as  follows : 

1  =.  1, 
7  =  7, 
10  =  1-8  +  0  =  eight. 
17  =  1-8  +  7  =  fifteen. 
20  r=  2-8  +  0  =  sixteen. 
56  =  5-8  +  0  =  forty-six. 
234  =  2-82  -f  3-8  +  4  =  one  hundred  fifty-six, 

etc. 
Let  us  take  the  arbitrary  number  z  as  the  base  of  the  scale. 
As  iu  our  scale  of  lO's  we  have 

234  =  2-102  +  3-10 -f  4, 
so  in  the  scale  of  z's,  the  digits  234  would  mean 

2^2  +  3;^  +  4. 
In  general,  the  combination  of  digits  abed  would  mean 

az^  +  hz^  +  cz  +  (L 


■il 


f  'I 


I 


A 


Divisibility  of  Xumbers  and  tlieir  Dij.vits. 

*^.*J5.  Theorem.    //  the  sum  of  the  dibits  of  any  niim 
her  he  suhtracted  froDi  the  niunhcr  itself,  the  remainder 
will  be  divisible  by  z  —  1. 


■    ;.j 


4- 


240 


DIVISIBILITY  OF  NUMBERS. 


;'l 

4 

,'♦ 

( 

i 

1 


Proiif.  Let  the  digits  be  «,  b,  c,  d.  The  number  expretjjed 
will  1^13  a;2;3  +  hz^  +  cz-\-d 

i5um  of  digits  =  a     -^  h    ■\-  c   -\-  d 

Subtriioting,     rem.  =  a{z^—\)  +  h{z^—l)  +  t'(,v'— I). 

Tlie  factors  z^  —  1,  2-  —  1,  and  z—  i  are  all  divisible  hv 
2—1  (§  lio).     Hence  the  theorem  is  proved.     (§  'I'-M.) 

'i'liEOiiEM.  In  any  scdie  liariiig  z  as  its  base,  iJte  sum 
of  tli6  (Uglts  of  any  niunber,  wJien  divided  by  z  —  1,  will 
leave  the  sanie  remainder  as  luill  the  miDiber  itself  icheit 
so  divided. 

If  we  put:      71,  the  number;     s,  the  sum  of  the  digits  ; 
r,  /•',  the  remainders  from  dividing  by  z  —  1; 
q,  q',  the  qnotients  ;    we  shall  have, 
Number,  n  =^   q  {z  —  1)  +  ?' 

Sum  of  digits,     6'  —  q'  {z  —  1)  +  r' 
Remainder,  {q  —  q'}  {z  —  1)  -\-  r  —  r'. 

Because  u  —  s  and  {q  —  q')  {z  —  1)  are  both  divisible  hy 
z  —  i,  their  dilTerence  /■  —  r'  must  be  so  divisible.  .Siner  r 
and  /•'  are  both  less  than  z  —  1,  this  remainder  can  be  dividid 
by  z  —  1  only  wlien  r  =  r',  which  proves  the  theorem. 

Zero  is  considered  divisible  by  all  numbers,  because  a  re- 
mainder 0  is  always  left. 

If  a  be  any  factor  of  z  —  1,  the  same  reasoning  will  api)ly 
to  it,  and  therefore  the  theorem  will  be  true  of  it. 

In  our  system  of  notation,  where  z  =  10,  the  above  thei»- 
rcms  may  be  put  in  the  following  well-known  form: 

Tf  tJie  sitni  of  the  digits  of  any  nuinherbe  divisitj/c 
by  />  ';;•  9,  the  nitmber  itself  ivill  be  so  divisible. 

These  are  the  onlv  numbers  of  which  the  theorem  is  true. 
because  3  is  the  only  divisor  of  9. 

Theorem.  If  from  any  n  umber  we  suhtraet  the  di'jH'^ 
of  the  even  poivers  of  z,  and  add  those  of  the  alter/fftfr 
powers,  the  result  will  be  divisible  by  z  -}•  \. 

Proof.    To  ay?  +  Iz'^  -\-  cz  +  d 

Add         a    —  b    -\-  c  —  d. 

Result,    a'(z^^^\y-rh{z^- 1  )~Vc{z'^. 


NUMBERS   AND    Til  EI  It   DIGITS. 


247 


The  factors  of  «,  h,  and  c  are  all  divisible  by  z-{-\  (g§  1)3, 

'j-i),  whence  the  result  itself  is  so  divisible. 

Applying  this  result  to  the  case  of  z  =■  10,  we  conclude: 
//'  on  Hubtrdcting  the  siun  of  tlir.  digits  in   the  pi  ace 

itf  units,  liundreds^  tens  of  tfionstaids,  etc.,  j'roiii  tin'  sinii 

iij'  the  (dtcrnate  ones,  the  renufinder  is  dirisil)Ie  tnj  II, 

t/ir  niondjer  itself  is  dirisHtlc  Itij  11. 

If  m  be  anv  factor  of  z,  it  will  divide  all  the  terms  of  the 

imniber 

az^  +  hz-  -[■  cz  -\-  d, 

except  the  last.  Hence,  if  it  divide  tiiis  last  also,  it  will  di- 
vide tiie  number  itself.  Applying  this  result  to  tiie  case  of 
:  =  10,  we  conclude: 

If  the  la*^  digit  of  any  niunher  is  divisible  by  a  fac- 
tor of  10,  the  niind)er  itself  is  divisible  by  tJutt  factor. 

The  factors  of  10  being  'i  and  5,  this  rule  is  true  of  these 
umubers  t)nly. 

It  will  be  remarked  that  if  the  base  of  the  system  iiad  been 
an  odd  nund)er,  we  could  not  have  distinguished  even  and  odd 
numbers  by  their  last  figure,  as  we  habitually  do. 

For  exam})le,  if  die  base  had  been  !),  th'  figures  72  would 
have  re])resented  what  we  call  sixtv-tive,  which  is  odd,  and  73 
wuuld  have  represented  wiiat  we  call  sixty-six,  which  is  even. 

The  use  of  the  base  10  makes  it  easy  to  detect  when  a  num- 
ber is  divisible  by  either  of  the  first  three  })rime  numbers,  2,3, 
and  5.  If  the  last  ligure  is  divisil)le  bv  2  or  5,  the  whole  num- 
lier  is  so  divisible.  To  ascertain  whether  3  is  a  factor,  we  find 
\vlietlier  the  sum  of  the  disi'its  is  divisible  bv  3. 

III  taking  the  sum,  it  is  not  nccopsary  to  includo  all  the  dibits,  but  in 
abiding  we  may  «miit  all  8'r  and  9's,  and  drop  3,  (>,  or  9  from  the  sum  as 
often  as  convenient.     Thus,  if  the  number  were 

92 1042 7 12, 
we  should  perform  the  operation  mentally,  thus: 

Drop  9  ;  3  +  1=3,  which  drop  ;  (),  drop  ;  4  +  2  =  0,  which  drop; 
T  +  1  =  8  +  3  :=  10,  which  leaves  a  remainder  1. 

EXERCISES. 

1.  Prove  that  if  an  even  number  leaves  a  remainder  1  when 
divided  by  3,  its  half  will  leave  a  remainder  2  when  so  divided. 


*■  i 


r  : 


i;  Jl 


'     I  T 
■    i 


r    < 


I'       I 


248  DIVL-iiniLITY   OF  NUMBERS. 

2.  If  from  any  number  we  sul)tract  the  sum  of  units'  digit 
j)lus  the  ])r()(luet  oi'  the  tens'  digit  by  i,  plus  the  product  of 
the  iiundreds'  digit  by  i^,  etc.,  the  remainder  will  be  divisible 
i)y  H)  —  /.      (/  may  be  any  integer,  ])ositive  or  negative.) 

NoTK.  When  i  —  1,  this  gives  the  rule  of  'J'.s  and  when  t  -  —  1,  tlic 
rule  of  11 'a. 

I'riiiKi  FiK'tor.s  of  Niiinbcrs. 

2.*UJ.  First  Fundamental  Thkouhm.  .4  product  can- 
not be  (livldcAl  by  a  prime  niunber  itiiless  one  of  the  few- 
tors  IS  divisible  by  that  prime  niunber. 

Note.  This  theorem  is  not  true  of  composite  divisors.  For  exam- 
]»le,  neitlier  8  nor  0  is  divisible  l)y  0,  but  the  product  8-9  =  72  is  so 
divisible.  But  if  we  take  as  many  numbers  as  we  jilease  not  divisible  liy 
7,  we  shall  always  find  their  product  to  leave  a  remainder  when  we  try 
to  divide  it  by  7. 

To  make  the  demonstration  better  understood,  we  shall  first  take  ii 
special  case : 

The  product  GGr?  is  not  divisible  by  7,  unless  a  is  divisibh' 

hj  7. 

Proof.     Sui)pose (50«  div.  by  T 

7  goes  into  GG  0  times  and  3  over,  because  7-9  =  G3,  iV.\a  div.  by  7 

Therefore,  l)y  Theorem  ll,%'Z'd\, 3rt  div.  ])y  7 

2 

3  goes  into  7  2  times  and  1  over,     ^lultiply  by  2,     (i«  div.  by  7 

Subtracting, 7«  div.  by  T 

We  have  left, .        ^^  div.  by  7 

Hence,  if  GGrt  is  divisible  by  7,  then  a  is  divisible  by  7. 

Gauss's  Demonstration.  If  it  be  possible,  let  am  be  the 
smallest  multiple  of  m  whicli  is  divisible  byy;,  when  neither  // 
nor  m  is  so  divisible.  If  a  is  greater  than  p,  then  let  p  go 
into  a  b  times  and  r  over,  so  that 

a  =  bp  -\-  r, 
or  a  —  bp  =  r. 

Then,  am  div.  by  p. 

Subtract  bpm    ''        " 

Remainder,  {a  —  bp)  ni    "'        " 
Or  rm    '•        " 


PRIME  FACTORS   OF  NUMBERS. 


240 


That  is,  if  am  is  divisible  hy  ;;,  so  is  ?•?//,  whore  r  is  less 
tliiin  p. 

Therefore  the  smallest  multiple  of  in  which  rullils  tlie  con- 
ditions must  be  less  than  pm. 

'Ihercl'ore,  let  a  <  p.     Let  (t  go  intoy^  c  (iiues  and  n  over, 

io  that 

p  —  ca  -\-  s, 

,,!•  p  —  ca  —  .v. 


Then 


pm  div.  by  p. 

cam.    ''        "     (by  hypothesis). 
« 


{p  —  ca)  m    " 


a 


Subtracting, 

Or,  sm 

Therefore,  s  being  less  than  a,  a  is  not  tlu'  smallest  mul(ii)lo; 
wlience  the  hypothesis  that  a  is  the  smallest  is  impossible. 

General  DemouHtration.     Suppose 

Pj  a  prime  number  ; 
a,  number  not  divisible  by  7;; 
am,  a  product  divisil)le  by  7;. 

We  have  to  prove  that  m  must  be  divisible  by  p. 

Let /J  go  into  rf  q  times.     Because  a  is  not  divisible  by^;, 
a  reiuainder  r  will  be  left.     That  is, 

a  ^  pq  +  /',     <>!'    «■  —  pq  =  ''. 

Let  r  go  into  p  q  times  and  leave 
11  remainder  ;•'.     Then, 

p  =  qr  ^  r', 

and  hccause  pm  and  q'rm  are  both  di- 
\isil»le  by  ^j,  rm  is  so  divisible. 

In  the  same  way,  if  r'  goes  into  p 
<j"  times,  and  leave  the  remainder  r", 
r'ln  will  be  divisible  hy  p.  Mince  each 
of  the  remainders  ;*,  r',  /•",  etc.,  must 
1)0  less  than  the  i)receding,  we  shall  at 
length  reach  a  remainder  1,  which  will  give 

711  divisible  by  p.        Q.  E.  D. 


am 

div. 

'^y  P' 

P^&. 

u 

.<i 

}'m 

a 

a 

q'rm 

a 

a 

pm 

a 

a 

r'm 

a 

a 

q"r'm 

a 

a 

pm 

a 

u 

r"m 

a 

« 

i    V 


H 


'H 


250 


nTVIsililLITY   OF  NlLVIiEUS. 


% 


s  . 


Jvxtcusion  to  Several  Fadors.  If  tn  is  a  product  h  x  n,  aiul 
h  is  not,  divisible  by  p,  tlien  we  may  show  in  the  same  way  tliut, 
n  must  be  so  divisible.  11'  n  ^  cs,  and  r  is  nat  divisible,  tlitu 
*'  must  be  divisible,  and  so  on  to  any  number  of  factors. 

Hence, 

Theorem,  //'ft  prodiwt  of  dini  nnnihrr  of  jncltivi^  is 
divisih/e  hij  a  prinir  niunhrr,  then  one  of  t/ie  factors 
must  be  dirlsibie  by  t/w  same  prime. 

This  theorem  is  the  loj^ical  equivalent  of  the  one  ju,-t 
enunciated  as  the  iirst  fundamental  theorem. 

Note.  Tlu*  atudont  will  remark  why  tlio  preceding  demonstratinti 
applies  only  when  th<!  divisor  ;?  is  a  prime  niiml)er.  If  !•  were  compo.-iic, 
^.e  mijrht  reach  a  reniiiindcr  which  would  exactly  divide  it,  uud  tlieii  llit- 
conclusion  would  not  follow, 

2'il,  Se(^oxi)  Fundamental  Theoiiem,  ./  nunilnr 
can  be  dii'ided  into  jnirie  factoids  in,  oiil if  one  way. 

For,  suppose  we  could  express  the  number  N  in  tlu;  two 

v:ays  (§  204,  Cor.  'Z), 

N  =  p"-  7^  i-y, 

N  =  (C^  b"  6"^, 

where 7?,  q,  r,  etc.,  a,  h,  r,  etc.,  are  all  prime  numbers.     Then 

If  common  prime  factors  ai>peared  on  both  sides  of  this 
ecptation,  we  could  divide  them  out,  leaving  an  equation  in 
which  the  prime  factors  p,  q,  ?•,  etc.,  are  all  different  from 
(I,  h,  c,  etc. 

Then,  because  a,  h,  c.  etc.,  are  all  prime,  none  of  them  ;iiv 
divisible  hyp.  Therefore,  by  the  first  fundamental  theorem. 
their  products  cannot  be  so  divisible.  But  the  left-hand  meiii- 
her  of  the  equation  is  divisible  by  p,  because  p  is  one  of  its 
factor,-:.     Therefore  the  equation  is  impossible. 

Rem.  This  theorem  forms  the  basis  of  the  theory  of  the 
divisibility  of  numbers. 

The  preceding  theorems  enable  us  to  place  tlie  definition 
of  numbers  prime  to  each  other  in  a  new  shape. 


BINOMIA  L    COEFFK  'lENTS. 


251 


I'monstration 


Two  numbers  are  said  to  be  prune  to  each  other 
when  they  have  no  common  prime  factors. 

Example.  If  one  number  i.s  p'^ffr'*,  and  the  other  is 
tif'b''c''  {p,  q,  r,  etc.,  and  a,  h,  c,  etc.,  bein<jf  ])rinic  nunihers), 
tlicn,  if;;,  <y,  r,  etc.,  are  all  ditferent  from  a,  by  r  (.  .,  the  two 
numbers  will  be  prime  to  each  other. 

Elementary  Theorems. 

238.  The  following  general  theorems  follow  from  the  t.vo 
piveeding  fundamental  theorems,  and  their  demonstration  is 
in  part  left  as  an  exercise  for  the  student. 

I.  Jfo  power  of  an  irrcO^'cihle  vulgar  fraction  can  he 
a  whole  nuniber. 

Note.  An  irreducible  'ilgi  Traction  is  one  which  is  re- 
duced to  its  lowest  terms 

II.  Corollary.  Jfo  roob  of  a  whole  number  can  he  a 
vulgar  fraction. 

III.  //  a  number  is  divisible  hi/  several  divisors,  all 
prime  to  each  other,  it  is  also  divisible  by  their  product. 

Cor.  To  prove  that  a  number  ^V  is  divisible  by  a  number 
B  =  p'^q^  ry,  it  is  sufficient  to  prove  that  it  is  divisible  sepa- 
rately by  p'^f  by  q^,  by  rv,  etc. 

Example.  If  a  number  is  divisible  separately  by  5,  8,  and 
1),  it  is  divisible  by  S-S-  9  =  3G0.  Hence,  to  prove  that  a  num- 
ber is  divisible  by  300,  it  is  sufficient  to  show  that  5,  8,  and  'J 
are  all  factors  of  it. 

IV.  If  the  numerator  and  denominator  of  a  vulgar 
fraction  have  no  common  prime  factors,  it  is  reduced  to 
its  lowest  terms. 

Binomial  Coefficients. 

239.  TJieorem.  The  product  of  any  n  consecutive 
numbers  is  divisible  by  the  product  of  the  numbers 
1- 2-3  ....  71,  or  n  ! 


I  %■ 


\'\ 


i  i)i 


i}ni 


UINOMIA  L    COEFFICIENrS. 


fltf: 


Hem.  The  theorem  implies  that  all  binomial  coefficients 
arc  whole  numbers,  becauso  they  are  (|Uotient8  formed  by  di- 
viding the  product  of  n  cousecutive  numl)ers  by  h\ 

Proof.  1.  We  have  lirst  to  tind  the  prime  factors  of  the 
product 

1.2. 3. 4. 50 n  =  n\ 

l)cginning  with  the  factor  l. 

I.  The  numbers  divisible  by  2  are  the  even  numbers  2,  4, 

'n 


G,  etc.,  to  n  or  n  —  1,  the  number  of  which  is 


2 


Note.     The  expression 


n 
2 


here  means  the  greatest  whoJi' 


n~  1 


mtmhcr  in    - ,  which  is  ^  itself  when  n  is  even,  and  — r 
2  2  2 

when  71  is  odd. 


The  quotients  of  the  division  are 

X)      liy      (i,      'X,      •     .     .      . 


n 
2 


71 

4 


Of  these  (quotients, 
second  set  of  quotients, 

X  J       fV^       tJ y        •      •      •       • 

The  next  set  of  quotients  will  be 

X,      /V,      ... 


are  divisible  by  2,  leaving  the      I    we  nh 


71 

4 


71 

8 


The  process  is  to  be  continued  until  we  have  no  even  num- 
bers left. 

Therefore,  if  we  put  a  for  the  number  of  times  that  the 
factor  2  enters  into  n !  we  have. 


«   =: 


n 
2 

4- 

71 

4 

+ 

71 

8 

+  etc. 


II.  The  numbers  in  the  series  71 !  containing  3  as  a  factor  are 

3,  G,  9,  12,  etc., 


liJNOMlA  L    COEFFICIENTS. 


25:3 


;  a  factor  are 


(.f  wliich  tlie  number  is     ..    •    The  quotients  obtained  by  di- 
viiliii*,'  them  by  3  are      L.'  J 


1     *>    n 

X  y        /Vy        tjy        •       •       •       • 


P 


J 


Of  these  quotients,  arc  again  divisible  l)y  3,  and  so 

(111  as  before.     Hence,  if  we  put  /i  for  the  number  of  times  n  I 
contains  3  as  a  factor,  wo  liavo 


»-[«]*  p] 


+ 


n 


+  etc. 


In  the  same  way,  if  k  be  any  prime  number,  n\  will  con- 
tain k  as  a  factor 


m 


n  j       I  n 

7.   I  +  I    ^, 


+ 


11 

k" 


+  etc.  times. 


Note.  This  elegant  process  enables  us  to  find  all  the  prime 
factors  of  nl  without  actually  computing  it,  and  thus  to  ex- 
hibit 71 !  as  a  product  of  prime  factors.  If  we  suppose  n  =  12, 
Wii  fehcill  tiud, 

12!  =  1.2.3 12  =  210.35.52.7.11. 

2.  Next  let  us  find  the  prime  factors  of  the  product 

(rt  +  1)  («  +  2) (a  +  n), 

wliich  contains  w  factors.  Dividing  successively  by  2,  3,  5,  7, 
etc.,  it  is  shown  in  the  same  way  as  before  that  the  prime  fac- 
tor j)  is  contained  in  the  product  at  lead 


+  etc.  times. 


whatever  prime  factor  /)  may  be.  Therefore  the  numerator 
((^4-1)  (a  +  2) . . . .  {a-\-n)  contains  all  the  prime  factors  found 
in  n\  to  at  least  the  same  power  with  which  they  enter  wl 
Hence  (§  238,  III),  the  numerator  is  divisilde  by  n\ 

Cor.    If  the  factor  a-\-7i  in   the   numerator   is  a  prime 
number,  that  prime  cannot  be  contained  in  til  because  it  is 


rw"i 

Fn  n 

+ 

A 

U^j 

»' 


1 

i  . 


•    ( 


254 


DIVIH0n8   OF  A    NU\fIiF:ii. 


^   ' 


(; 


greater  than  n.    Ilencc  tho  })iaonuul  factor  will  be  divisilile 
by  it. 

„  5«(i' 7  .,...,,    ,     „ 

ExAMi'LK.     ,  ,,  .,  IS  uivisil)k' by  7. 

We  may  .sliow  in  the  same  way  that  the  l)iii()niial  coellicicnf 
is  divisible  l)y  all  the  prime  numbers  in  its  numerator  whi(  h 
exceed  n. 

DiviHorH  of  ii  Number. 

240.  Def.    The  expression 

(P{m) 

is  used  to  express  how  many  numbers  not  greater  than 
•m  are  i)rime  to  iii. 

Example.    Let  us  find  the  value  of  </)(0). 

1  is  j)rime  to  9,  because  their  G.  C.  D.  is  1. 

Q        «  «  (f  U  «  U 

3  is  not  prime  to  9,  because  their  0.  C.  D.  is  3. 

4  is  prime  to  9. 

5  "  " 

6  is  not,  because  G  and  9  have  tho  0.  C.  D.  3. 

7  is. 

8  is. 

9  is  not. 

Therefore,  th^  numbers  less  than  9  and  prime  to  it  are 

1,  2,  4,  5,  7,  8, 
which  are  six  in  number.    Hence, 

0  (9)  =  6. 

The  numbers  less  than  13  and  prime  to  12  are  1,  5,  '«•  H- 

Hence, 

0(12)  =  4. 

We  find  in  this  way, 

0(1)  =1,  0(2)  =  1,  0(3)  =  2, 

0(4)  =2,  0(5)  =4,  0(0)  =  2, 

0(7)  =  6,  etc.,  etc. 


DIVISORS   OF   A    NUMBKll 


255 


Cor.  1.  The  numbt'i  I  in  prime  to  itself,  but  no  other 
number  is  prime  to  itself. 

Cor.  2.     If  m  be  a  prime  num1)er,  then 

0(m)  =  m  —  1, 

licciuise  the  numbers   1,  2, 13, ....  m  —  i    are  tlien  all  prime 

t(»  ///. 

The  following  remarkable  theorem  is  assoeiated  with  the 
functions  <!>  (///). 

*Z-i\,  Theorem.  If  iV  bo  any  nnmhcr,  and  c7,,  r/.^, 
^/g,  etc.,  all  its  divisors,  unity  andxVinclud(^d,  then 

0(^i)  4-  0(^2)  +  'Al^^a)  +  ^'tc.  =  N. 

Example.    Tjct  the  number  be  18. 

The  divisors  are  1,  2,  3,  G,  9,  18.    We  find,  by  counting, 

0(1)  =  1 
0(2)  =  1 
0(3)  =  2 
0(1;)  =  2 
0(9)  =  0 
0(18)  =  ^ 

Sum,     18. 

To  show  how  this  comes  about,  write  down  the  numbers 
1  to  18,  and  under  each  write  the  greatest  common  divisor  of 
that  number  and  18.     Thus, 

Num.,    1  2  3  4  5  G  7  8  9  10  11  12  13  U  15  IG  17  18. 
G.C.D.,  12321G12921G1232118. 

Necessarily  the  numbers  in  the  second  lino  are  all  divisors 
of  18  as  well  as  of  the  numbers  over  them. 

The  divisor  1  is  under  all  the  numbers  prime  to  18,  so 
that  there  are 

0(18)  =  div.'sors  1. 

If  n  be  any  number  over  tho  d" visor  2,  then  -  and  — ,  or 
!»,  must  be  prime  to  each  other.     (§  232,  Cor.  1.)    That  is,  the 


'f 


!         I 


1,1 
Ml 


■!aPF^!5es-.. ,,  j.iL,-.  iji-.-iML!,. 


250 


DIVISORS  OF  A    NUMBER. 


I  _ 


»»■ 


lit    » 


M 


numbers  w  arc  all  those  which,  when  divided  by  2,  are  prime 
to  9..     So  there  arc 

0(0)  divisors  2. 

The  divisor  3  marks  all  numbers  which,  when  divided  by  3, 

18 
are  prime  to  —  =  G.     Hence,  there  are 
o 

(p  (0)  divisors  3. 

In  the  same  way  there  are  <p  (3)  divisors  G,  0(2)  divisors !), 
and  0(1)  divisor  18. 

The  total  number  of  these  divisors  is  both  18  and  0(18) 
-}-  0  (9)  +  etc.     Hence, 

0(18)  +  0(9)  +0(G)-f-0(3)  +  0(2)  +0(1)  =  18. 

General  Proof.     Let  m  be  the  given  number; 
fZj,  f/g'  ^^35  ^'tc,  its  divisors; 
7i'   72'   '73'  the  quotients    .   ,     .- ,  etc. 

"l        "2 

The  quotients  (7,,  5*2,  etc.,  will  be  the  same  numbers  as  r/,. 
c?2.  etc.,  only  in  reverse  order.  The  smallest  of  each  row  will 
be  1  and  the  greatest  m.     We  shall  then  have 

m  =  f/i  <7i  —  r/g  72  =  ^^3  '73'     etc. 

From  the  list  of  numbers  1,  2,  3,  ....  ?/?,  select  all  tho^^o 
which  have  r/,  (unity)  as  the  greatest  common  divisor  with  ;/^ 
then  those  which  have  (/^  as  such  common  divisor,  then  those 
which  have  r/3,  etc.,  till  we  reach  the  last  divisor,  which  will 
be  ???  itself,  and  which  will  corresi)ond  to  tn. 

The  numbers  having  unity  as  G.  ('.  D.  will  be  those  prime 
to  in,  by  definition.     Their  number  is  0(wO- 

Those  having  (L  as  G.  C.  D.  with  ?//  will,  when  divided  bv 

CD  1^  '  *■ 

^7g,  give  quotients  prime  to    j-  or  to  q^.     Moreover,  such  (iiu)- 

tients  will  include  all  the  numbers  not  greater  than  r/j,  ami 
prime  to  it,  because  each  of  these  numbers,  when  multipliid 
by  r/g,  will  give  a  number  not  greater  than  >«,  and  having  d» 
as  its  G.  C.  D.  with  m.     Hence  the  number  of  numbers  not 


FEHMAT'S    THEOREM. 


257 


greater  than  ???,  and  lKivin,i:f  lU  as  its  G.  C.  D.  with  m  will 

1.0  0(^2). 

Continuing  the  process,  we  shall  reach  the  divisor  m,  which 
will  have  m  itself  as  G.  C.  D.,  and  which  will  count  as  the 
iiiinilx^r  corresponding  to  </>  (1)  =  1  in  the  list. 

The  m  numbers  1,  ^,  3, . .  .  .  m  are  therefore  ec^ual  in  num- 
ber to 

0O'O  +  0(V2)  +  0(73)4-  ....  +0(1); 

or.  since  the  ((uotients  and  divisors  are  the  same,  only  in  re- 
verse order,  we  shall  have 

0(1)  -f  0(^/3)  -f  qi{d^)  +  ....+  0(/»)  =  w. 

24'^.  Fkumat's  Tiieoukm.  If  p  he  ait//  /trijne  nurn- 
hn\  and  a  ha  a  luuiibcr  prima  to  p.  Ilirii  tiJ'-^  —  1  will  be 
divisible  by  p. 

Examples,     n'^  —  1  is  divisible  by  5  ;  a^  —  1  is  divisible  by  7. 

Proof.  Develop  ai'  in  the  following  way  by  the  binomial 
theorem, 


a* 


[i  +  («-i)F 


=  1  -\-p{a  -  1)  -f-  [{^  (.,_  1)2  +  ....  +  {a  -  If. 
Because  p  is  prime,  all  the  binomial  coelTicients, 


P> 


2   ' 


etc.,     to 


(.-^-.> 


are  divisible  by  p  (§  '230,  Cor.).     Transposing  the  terms  of  the 

last  member  of  the  eijuation  which  are  not  divisible  by  ^>,  wc 

liiul 

a^'  —  (a  —  1)''  —  1  =  a  multij)le  of;). 

or      a^  —  a  —  [(«  —  1)^'  —  {a  —  1)]  =  a  multiple  of  p. 

Supposing  X  =  2,  this  equation  shows  that  'Z^*  —  2  is  a 
multiple  of  />;  then,  sn])])osing  a;  =  3,  wo  show  by  §  )i'M, 
Til.  H,  that  ',]''  —  .3  is  such  a  multiple,  and  so  on,  indelinitely. 

Hence,  a**  —  a  =  a  multiple  of  p. 

whatever  be  a.     But  a''  —  a  ~  {aP"^  —  l)a,  and  because  this 
l>roduct  is  divisible  by  p,  one  of  its  factors  must  be  so  divisible 
(§  23G).     Hence,  if  a  is  prime  to^,  a^^~*  —  1  is  divisible  by  7;. 
17 


M- 


r 
i     t 


»* 


258 


CONTINUED   FRACTIONS. 


r, 


CHAPTER     II. 

OF     CONTINUED      FRACTIONS. 

^43.  Any  proper  fraction  may  be  represented  in  the  form 
— ,  where  x^  is  greater  than  unity,  but  is  not  necessarily  a  whole 
number.    If  a^  be  the  greatest  whole  number  m  x.^,  we  can  put 

^1  ==  ffi  +--, 

■where  ^g  will  be  greater  than  nnity.     In  the  same   .ay  wc 

may  put 

1 

2^8   =  fl'a  H ' 


2^3    =    ^3    + 


etc. 


X, 

etc. 


If  for  each  rr  wo  substitute  its  exi  _ssion,  the  fraction 
"will  take  the  form 

J^ I _J 


^h  + 


■Vi 


«i  + 


rto  H ■ ,  (Jtc,  etc. 


X 


If  the  substitutions  are  continued   indefinitelv,  the  form 
will  be  2 


«i  + 


a.,  + 


1 


^3    +       - 


''-  +  7u 


Such  an  expression  '  ■>  called  a  continued  fraction. 

Dcf.  A  Continued  Fraction  is  one  of  which  the 
denominator  is  a  whole  number  plus  a  fraction ;  the 
denominator  of  this  last  fraction  a  wholo  number  plus 
a  fraction,  etc. 


CONTINUED    FRACTIONS. 


259 


A  continued  fraction  may  cither  torniinate  with  one  of  its 
di'Hominators  or  it  may  extend  indefinitely. 

Def.    When  the  number  of  quotients  a  is  finite,  the 
fraction  is  said  to  be  Terminating. 

344.  Problem.     To  find  the  value  of  a  continued 
fraction. 

We  first  find  the  vaUie  when  we  stop  at  the  first  denomina- 
tor, then  at  the  second,  then  at  tlie  third,  etc. 

Using  only  two  denominators,  the  fraction  will  be 


X, 


1 


^1  +  -- 


a^r^  +  1* 


F  l)eing  put  for  the  true  value  of  the  fraction. 

To  find  the  expression  with  three  terms,  we  put,  in  the 
preceding  expression,  a^  -\ in  place  of  x^.    This  gives 


«2    +   — 


F=: 


X 


3  


a.x^  +  1 


a. 


f'l^z  +  ^  +  1 


(a^a^  +  l);'-3  +«i 


X, 


To  find  the  result  with  the  fourth  denominator,  we  8u))sti- 
tute  x^  =  a^  -\- 


X, 


F 


The  fraction  becomes ; 
(«2^3  +1)^4  +  ^2 


[(r/j^g  +  1)  «3  +  ^1 J  a^4  +  «i«2  +  1 


{<') 


To  investigate  the  general  law  according   to  which   the 
{Successive  expressions  proceeil,  we  put 

P,  the  coefliicient  of  x  in  any  numerator; 
F,  the  coefficient  of  x  in  the  denominator; 

Q,  the  terms  not  multiplied  by  x  in  the  numerator  ; 
Q',  the  terms  not  multiplied  by  x  in  the  denominator ; 

and  we  distinguish  the  various  expressions  by  giving  each  P 
and  Q  the  same  index  as  the  x  to  which  it:  belongs. 


i  '  I 


,>r'i 


1>60 


CONTINUED   FRACTIONS. 


♦» 


(^) 


Tlien  \vc  may  represent  each  value  of  F\n  the  form, 

~  P'  0'  -I-  (/' 

where  /  may  take  any  vahie  necessary  to  distinguish  the  frac- 
tion.    Comparing  with  the  fractious  as  written,  we  see  that : 

p^  =  i,  Q,  =  o,    p;  ==-.«,,  (>2  =  i;     (') 

P3  =:  f'2>        Q3  =  ^   ^'a  --  ('tf(2-^^^    Q\  =  «i ; 

To  show  tliat  this  form  will  r-ontinuc,  how  far  soever  wi 
carry  the  comi)utation,  we  put  in  the  expression  {b)  the  geneml 
value  of  Xi, 


Xi  =  Hi  -\ , 

Xi+\ 


which  gives,         F 


(«i  Pi  +  Vi)  3*if3  4-  Pi 


(•/) 


To  show  the  general  law  of  succession  of  the  terms,  let  ii< 

comi)are  the  general  equatior-  pj)  with  (d).    Putting  i+1  lor 

i  in  {b),  it  l)eeomes, 

P.  .-r,  .  _L  n.  . 


~"  P'     r       4-0' 

^t+r  ill  ^  '»^ifi 


CcMiparhig  this  with  (r/),  we  find 

Ai  =z  aiPi+  Qi, 

whence,  Qi  =  Pi-\. 

Substituting  this  value  of  Qi  in  the  equation  previous,  it 
bectjmes 

P,+,  =  fti  Pi  +  Pi-i.  (0 

Working  in  the  same  way  with  the  denominators,  we  find 

P;,,  =  a.  P; +  /-„,.  (;/) 

By  supposing  i  to  take  in  succession  the  values  1,  "Z,  3,  Qic, 


CONTIN  UED    FRA  CTIONS. 


261 


llicse  fonnulae  show  that  the  successive  vahies  of  P  may  bo 
Lomi)uted  thus : 

P,  =.0, 


P2  =  h 


(from  c)  ; 


Also, 


P,  =  a,P,  +  i'3, 

etc.,  to  any  extent. 
P'    —  1 

p;  =  n,p'^  +  p;, 

P'    —  //    /*'    a-  />' 
etc.  etc. 


Since  each  vahio  of  Q  is  equal  to  the  value  of  P  havinf^  the 
next  smaller  index,  it  is  not  necessary  to  compute  th<;  ^/s  sep- 
arately. 

If  the  fraction  terminates  at  the  n'^  value  of  (u  we  shall 
liave 

Xn  =■  an,  exactly. 

If  it  does  not  terminate,  we  have  to  ne^lri  i  ail  ihe  denom- 
inators after  a  certain  point;  and  calling  '\v)  last  denominator 
we  use  the  n^,  we  must  suppose 

In  either  case,  the  expr  am  {b)  will  give  tlie  value  of  tho 
fraction  with  which  we  stoi   by  putting  i  =  n  and  3'n  —  «,»• 


Therefore, 


yp        fin  Pn  H-   Qn 

^     ~  W ^'  ' 

an  Pn  +   Qn 


or,  substituting  for  Q^  and  Q'^  their  vahses  in  (^), 

anPn  +  Pn-\ 

But  the  general  expressions  (/)  and  {(j)  give 


■(T 


t'» 


262 


Therefore, 


CONTINUED    FRACTIONS. 

dn  Pn  +  Pn-\  =  J'n+1- 

Pn'r\ 


F  = 


Tlierefore,  to  find  the  value  of  the  fraction  to  the  n'^^ 
term,  we  have  only  to  compute  the  values  of  Pnn  and 
Pnh  ifithout  taking  any  account  of  Q. 

Example.     Take  tlic  fraction, 


1+- 


2  + 


3  + 


4  + 


1 


5  etc. 
Here,    a^  =  1,    a.^  =2,    a.^  =  3,  .  .  .  .  m  =:  i. 

We  now  have,  by  continuing  the  fomiulse  {c)  and  (/),  and 
nsing  thosf?  vuhies  of  a^,  a^,  etc.: 

P,  =  0, 

P2  =  h 

P^  r=  a^P^  -h  Pi  =  a^  =  2, 

P4  -^«3^3  +  Pg  =3.2  +  1  =  7, 

P,  =  a,P,  -h  Pi  =  4.7  +  2  =  30, 

P,  ^  a,P,  +  P4  =  5-30  +  7  =  157, 

etc.  etc.  etc. 

p'jj  = «,  =  1, 

p;  =:a^p'^  +  p;  =  v.i  +  1  =  3, 

K  =  '^3/*s  +  p;  =  3.3  +  1  =  10, 
/>;  =  a^p\  +  p;  =  4.10  +  3  .=  43, 
p;  rr  rt^r;  +  p;  =  5.43  +  10  =  aafi. 

Therefore,  supposing  in  succesaioii,  n  =  1,  «  =  2,  «  =  3, 
etc.,  we  have,  for  the  successive  approximate  values  of  the 
fraction. 


CON  TIN  UED    FJiA  CTIONS. 


263 


For  71  =  1, 
For  71  =  2, 

For  n  =  5, 


p 

2 


■*  2    —    '/>' 

3 

P 
P      —    ^    ^ 

6 


2 

3 

157 

225' 


Tliege  successive  approximate  values  of  the  continued  frac- 
lioii  are  called  Converging  Fractions,  or  Convergents. 

345.  The  forms  (/)  and  {(j)  may  he  expressed  in  words  as 
follows: 

The  iiuvicrntor  of  each  eonvct'^cnt  is  formed  hij  mul- 
ti plying  the  preceding  numerator  hi/  the  corresponding 
a,  and  adding  the  second  numerator  preceding  to  the 
product. 

The  successive  denominators  are  formed  in  the  same  way. 

Example.  Tlie  ratio  of  the  motions  of  the  sun  and  moon 
relative  to  the  moon's  node  is  given  by  the  continued  fraction: 

\ 

124-i-. 


1  + 


2  + 


1  + 


4-fi- 


3  -f  etc. 

TiCt  us  find  the  successive  convergents.     We  put  the  de- 
jiominators  a^  —  12,  a^  —  1,  etc.,  in  a  line,  thus: 

a     r=    12,      1,       2,       1,       4,        3. 
i'     12         ~       ~ 


P'    = 


13'     38'     51'     242'     777 


0 


Under  n^  we  write  the  fraction  . ,  which  is  always  the  one 
with  which  to  start,  because  P^  =  0  and  P'l  =  1  (§  244,  c). 
Next  to  the  right  is  — ,  because  P^  —  \  and  P'g  =  n.  After 
this,  we  multiply  each  term  by  the  multiplier  a  above  it,  and 


t  • 


TT 


'¥^' 


264 


CONTINUED    FRACrrONS. 


rv 


add  tho  term  to  the   loft   to  o])tain    the  term  on  the  riglit. 
Thus,     a-l  -f  1  =  3,    '^.13  +  1-^  ==  38,    etc. 

Ex.  2.    To  compute  the  convergents  of 

1 


2  + 


1 


4  + 


1 


5i  + 


1 


4  etc. 

^     =     2,     4,     2,      4,       2, 

Numerators,        0      14       0       40 
Denomimitors,     i'    2'    9'    20'     8U'     198' 


4,       etc. 

89 
;^. ,     etc. 


EXERCISES. 


Reduce  the  following  continued  fractions  to  vulgar  frac- 
tions : 

1 


I. 


2. 


3  + 


7  +  J-. 
^  16 


3  +  1- 


Ji  + 


3  +  ^ 


3  + 


1  + 


3  + 


3  + 


5- 


«  + 


^  +  .- 


^'4- 


1  + 


1 


246.  Problem.     To  express  a  fractional  quantity  n^ 
a  continued  fraction. 

Let  K  be  the  given  fraction,  less  than  unity.     Compute  .r, 

fiom  the  formula, 

1 

Let  a^   be  the  whole  number  and  E'  the  fraction  of  x^. 

Then  compute 

_   1 

^8  -  j^r 


CONTINUED    FRACTT0N8. 


265 


Tx^t  r/g  be  tho  whole  nnmbor  and  R"  the  fraefion  of  rg. 
\Vc  continue  this  process  to  any  extent,  unless  some  value 
of  X  comes  out  a  whole  number,  wiien  we  stop. 

T.  T,  '-in  .       ,  „       . 

±iXAMPLE.    ±!iXprcss  ^^  as  a  contiuueu  fraction. 


_  1    _  73  _  ^   ,   21 

^'  -  7^  -  20  -  ^  "^  20  ' 

-  1  _  ??  -  1  j.i 

^^  ~  W  ~  21  ~  ^  "^21' 


.    _     1_  _  211  _  1 


It 


^A      7-,./, 


5' 


5 


-5; 


*  ~  R"  ""   1 

So  the  continued  fraction  is 

1 


.•.  rt,  =  2 


.-.    «2    =    1 


«3   =  4 


^4  =  5 


72'  = 
R'  = 
R'"  = 
i?'  =  0. 


21 

20" 

21' 
1 


2  + 


1  + 


1 


4  +  i. 


It  will  be  seen  that  the  process  is  the  same  as  that  of  find- 
ing the  greatest  common  divisor  of  two  numbers. 

EXERCISES. 

Develop  the  following  quotients  as  continued  fractions: 


I. 


113 
355' 


2. 


1040 
3^320' 


028 
925' 


247.  The  most  simple  continued  fraction  is  that  arising 
from  the  geometric  problem  of  cutting  a  line  in  extreme  and 
mean  ratio.     The  corresponding  numerical  proV)lem  is: 

To  divide  unity  into  two  such  fractions  that  the  less 
shall  he  to  the  greater  as  the  greater  is  to  unity. 

Let  r  be  the  greater  fraction.  Then  1  —  r  will  be  the 
lesser  one.    We  must  then  have 

\  —  r  ',  r    : :    r  :  1, 


206 


CONTINUED   FRACTIONS. 


wliifh  givos 

r»  =  1  —  r, 

or 

r»  +  r  _  1, 

or 

r(r -fl)  =  1, 

f\r 

1 

r  —  . 

l-fr 
Now,  let  us  put  for  /*  iu  the  lust  dciiomiiuitor  the  oxprcssion 


1  +  r 


-,  and  repeat  the  process  indofinitely.     We  sliall  liuve, 


1 


\f         * 


1  4- 


l  + 


1  + 


1  etc.,  nd  iufinilum. 

Now  we  may  form  tlie  successive  convergents  which 
approximate  to  the  true  vahio  hy  the  rule.  A.s  a'l  the  denom- 
inators n  are  1,  we  have  no  multiplyinfr*  hut  only  add  each 
term  to  the  preceding  one  to  obtain  the  following  one.  Thus 
we  find: 

0     1      1      2     3      5      ^      13      21      34 

i'    1'    2'    3'    5'    8'    13'    21'    34'    55'    ^'^^* 

The  true  value  of  r  may  he  found  hy  solving  the  quadratic, 


which  gives 


r  = 


2 


The  positive  root,  with  which  alone  we  are  \jOboenied,  is 

-1  +  V5 


r  = 


2 


=  0.61803390. 


The  values  of  the  first  nine  convergents,  with  their  errors. 


are: 


1:1=  1.0,  error  =  +  0.382. 

1:2=  0.5,  "          -0.118. 

2:3=  0.666....,  «           +0.0486. 

3:5=  0.600,  "          -0.0180. 

5:8=  0.G25,  «           -}- 0.00697. 


CONTINUKI)    Fit  ACT  IONS. 


i>07 


8  :  13  =  0.01538...., 

om>r  = 

—  0.0(K»r,5. 

13  :  ^1  =  O.OlliOL...  , 

.ft 

+  0.00101. 

21  :  34  =  O.OKIU:...., 

«< 

—  0.0(10307. 

34  :  55  =  0.()1«182...., 

t» 

4-  0.000 148. 

t'tf.                        I'tc. 

I'tc. 

'ir  errors. 


Kdatioiis  of  Suocossive  Convorf^oiits. 

^48,   TiiKoiiKM    I.      The    sitcrrssiue    convcvgcnts    (ire 
(i\  tenuitcl  11  too  birge  and  too  sin  (ill. 

Proof.     The   first   convergent   is   —     Tlie   true  dcnom- 


(t^ 


1 


iiiiitor  beinp;  n^  -\ ,  the  denominator  r?,  is  too  finniU,  and 

therefore  the  fraction  is  too  large. 

In  forming  (lio  second  fraction,  we  put        instead  of 


a. 


J\ 


Because  a^  <  .Tg,  this  fraction  is  too  hirge,  whicli  makes  the 
denominator  a,  H too  small. 

The  tliird  denominator  a^  is  too  small,  which  will  make 
the  preceding  one  too  large,  the  next  preceding  too  small,  and 
so  on  alternately. 

Theorem  II.    //  —  (uul  —-,  he  (tny  two  consecutive 
'     n  n  ^ 

convcrgcnts,  then 

mil  —  m'n  =  ±  1. 

Proof.    We  show  : 

(rt)  That  the  theorem  is  true  of  the  first  pair  of  convergents. 

(i3)  That  if  true  of  any  pair,  it  will  be  true  of  the  pair  next 
following. 

(«)  Tlie  first  pair  of  convergents  are 

1  1  «9 


a. 


«'+r 


1         a^a^  +  1' 


which  gives  7nn'  —  m'n  =  1,  thus  proving  («). 


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268 


CONTIX  UED    FRA  CTIONH. 


j> 


m      m       m' 


'i 


v 


I 


(/3)  Let 

be  three  consecutive  con\  orgcnts,  in  which 

mn'  —  m'n  =  ±  1.  (l) 

By  (/)  and  [g)  we  shall  have 

m"  =  am'  +  m, 
n"  ■=  an'  +  n. 

Multiplying  the  second  equation  by  m!  and  subtracting  the 
product  of  the  first  by  n',  we  have 

m'n"  —  m"n'  =  m'71  —  mn', 

which  is  the  negative  of  (1),  showing  that  the  result  is  ^  1. 

The  theorem  being  true  of  the  first  and  second  fractions, 
must  therefore  be  true  of  the  second  and  third  ;  therefore  of 
the  third  and  fourth,  and  so  on  indefinitely. 

Corollaries.    Dividing  (1)  by  nn',  we  have 

m      m!         .     1         T, 

;  =  ±  — )•      Fence, 

n        n  nn 

I.  Tlie  difference  between  the  tivo  successive  conver^- 
ents  is  equal  to  unity  divided  by  the  product  of  tJic 
denominators. 

Becaase  the  denominator  of  each  fraction  is  greater  than 
that  of  the  preceding  one,  we  conclude : 

II.  Tlie  difference  between  two  consecutive  convergenis 
constantly  diminishes. 

Combining  these  conclusions  with  Th.  I,  we  conclude  : 

III.  Each  value  of  a  convergent  always  lies  between 
the  values  of  the  two  preceding  convergents. 

For  if  R^,  R^,  R^  be  three  such  fractions,  and  if  R^  is 
greater  than  R^,  then  R^  will  be  less  than  R^.  But  it  must 
be  greater  than  R^,  else  we  should  not  have  Rg  —  Rq  numer- 
ically less  than  R^  —  R^.  Hence,  if  we  arrange  the  successive 
convergents  in  a  line  in  the  order  of  magnitude,  their  order 
wall  be  as  follows : 

-ft4j     Jig?    Rs'    ....     /tj,,    it^j    Ag, 

each  convergent  coming  nearer  a  true  central  value.    Hence, 


reater  than 


CON  TIN  UED    FRA  CTIONS. 


26Q 


IV.  Tlie  true  value  of  the  continued  fraction  nl- 
irays  lies  between  the  values  of  two  consecutive  con- 
vergents. 

Comparing  with  (I),  we  conclude  : 

V.  Tfie  eiTor  which  we  make  by  stopping  at  mni  con- 
vergent can  never  be  greater  than  unity  divided  Inj  f/w 
product  of  the  denominators  of  that  convergent  and,  the 
one  next  following. 


EXAMPLE. 


Referring  to  the  table  of  values  of  ^{V5  —  1)  in 
we  see  that : 


247, 


Error  of    2  :  3  <  ^-- ; 

o  •  0 

(for  .0486  <  pV 

Error  of    3  :  5  <  „"^_  ; 

0"  O 

etc. 

(for  .018    <  A), 
etc. 

Hence,  in  general,  continued  fractions  give  a  very  rapid 
approximation  to  the  true  value  of  a  quantity.  Their  princi- 
pal use  arises  from  their  giving  approximate  values  of  irrational 
numbers  by  vulgar  fractions  with  the  smallest  terms. 

EXAMPLE. 

Develop  the  fractional  part  of  a/2  as  a  continued  fraction, 
and  find  the  values  of  eight  convergents. 

Because  1  is  the  greatest  whole  number  in  \/2,  we  put 

V2  =  l  +  i; 


X 


(1) 


whence, 


X    =2 


a/2-1 

Rationalizing  the  denominator,  §  185, 

X  =  •v/2  +  1. 
Substituting  for  \^2  its  value  in  (1), 

1 


X  =  2  -{- 


i-»:i 


■f. 


H 


'I 

I 


■I  ; 


,1  >  I 


,P:I 


X 


270 


CONTINUED   FRACTIONS. 


;!; 


i    ♦ 


Putting  this  value  of  x  in  (1)  and  again  in  the  denominator, 
and  repeating  the  substitution  indefinitely,  we  find 

Va  =  1  +  ^- T 


2  + 


2  +  —^ 
^  "^  2  etc. 

Forming  the  convergents,  we  find  them  to  be 

1       2       5       13      29       70       169      408 
2'     5'     12'    29'     70'     169'    408'     985' 


etc. 


Adding  unity  to  each  of  them,  we  find  the  approximate 
values  of  a/2  : 


?      !      IZ      ^      ??      i??      5-^      1??_^ 
2'     5'     12'    29'     70'     169'    408'      985~' 


etc. 


Rem.  The  square  root  of  2  may  be  employed  in  finding  a 
right  angle,  because  a  right  angle  (by  Geometry)  can  be  formed 
by  three  pieces  of  lengths  proportional  to  1,  1,  'v/2.  If  avc 
make  the  lengths  12,  12,  17,  the  error  will,  by  Cor.  V,  be  less 

than  r^r-TTi,  or  less  than  -—  of  the  whole  length. 
12,29  o48 

EXERCISES, 

Develop  the  following  square  roots  as  continued  fractions, 
and  find  six  or  more  of  the  partial  fractions  approximating  to 
each : 

I.    Vs.  2.    Vs.         3.    Vg.         4.    VlO. 

5.  Develop  a  root  of  the  quadratic  equation 


x^ 


ax  —  1 


0, 


commencing  the  operation  by  dividing  the  equation  by  x. 

Periodic  Contiiiued  Fractions. 

349.  Def.  A  Periodic  continued  fraction  is  one  in 
which  the  denominators  repeat  themselves  in  regular 
order. 


CONTINUED    FRACTIONS. 


271 


Example,    A  continued  fraction  in  which  the  succe&sive 
denominators  are 

2,    3,    5,    2,    3,    5,    2,    3,    5,    etc.,    rtfZ  infinitum, 

is  periodic. 

A  periodic  continued  fraction  can  be  expressed  as 
the  root  of  a  quadratic  equation. 

EXAMPLES, 


1  + 


2  + 


l  +  o 


2  +  etc. 

If  we  put  X  for  the  value  of  this  fraction,  we  have 

1 


X  = 


1  + 


We  find  the  value  thus : 

1,        2  +  x. 

0  1 

1'  1' 


2  -f-  ic 


2  -\-  X 


3  +  x 

Because  this  expression  is  x  itself,  we  have 

_  2  +  X 
^~3+x' 

which  reduces  to  the  quadratic  equation 

x^  +  2x  =  2. 

2.  Let  us  take  the  fraction  of  which  the  successive  denom- 
inators are  2,  3,  5,  2,  3,  5,  etc.,  namely, 

1 


X  = 


2  + 


3  + 


5  + 


2   +  7 


3  -f  etc., 


^  ,n 


♦  i 

II; 


t'J 


-      I    !    i 


I  ,i     f 


272 


CONTINUED   Fli ACTIONS. 


i 


■    i 


♦ 


1».- 


or, 


X  = 


2  + 


3  + 


We  compute  thus : 

2,  3, 

0  1 

1'  2' 


5  +  a; 


a;  +  5. 
3 


Sx-\-  16 


7a; +  37 

Hence  we  hove,  to  determine  x,  tlie  quadratic  equation, 
3a;  +  16 


X  = 


7a; +  37' 


or 


7a;2  +  34a;  =  16. 


350.  Development  of  the  Root  of  a  Quadratic  Equation, 
A  root  of  a  quadratic  equation  may  be  developed  in  a  continued 
fraction  by  the  following  process.  Let  the  equation  in  its 
normal  form  be  (§  192), 

mx^  +  wa;  +  JO  =  0,  (1) 

771,  n,  liud  j9  being  whole  numbers.     We  shall  then  have 


a; 


—  n  ±  Vw/^  —  ^Tnp 


2ni 


Let  a  be  the  greatest  whole  number  in  x,  which  we  may 
find  either  by  trial  in  (1)  or  by  this  value  of  x.    Then  assume 

1 

a;  =  a  +  — , 

and  substitute  this  value  of  x  in  the  original  equation.  Then, 
regarding  x^  as  the  unknown  quantity,  we  reduce  to  the  nor- 
mal form,  which  gives 

{ma^  +  na  -\-  p)x^^  +  {2ma  +  w)  a^i  +  w  =  0.         (2) 
If  a^  is  the  greatest  whole  number  in  x^,  we  put 

and  by  substituting  this  value  of  x^  in  (2),  we  form  an  equa- 
tion in  Xq.  Continuing  the  transformations,  we  find  the 
greatest  whole  number  in  x^,  which  will  be  called  ^g?  ^^^  so  on. 
The  root  will  then  be  expressed  as  a  whole  number  a  phis 
the  continued  fraction  of  which  the  denominators  are  a^,  «jj, 


BOOK    X. 
THE     COMBINATORY    ANALYSIS, 


CHAPTER     I. 

PE  R  M  U  TATI  0  N  S. 

351.  Def.  The  different  orders  in  which  ta  number 
of  things  can  be  arranged  are  called  their  Permuta- 
tions. 

Examples.    The  permutations  of  the  letters  a,  h,  are 

ah,     ha. 
The  permutations  of  the  numbers  1,  2,  and  3  arc 
123,     132,     213,     231,     312,     321. 

Problem.  To  find  Jiow  many  permutations  of  any 
^Iven  number  of  things  are  possible. 

Let  us  put 

Pi,  the  number  of  permutations  of  i  tilings. 

It  is  evident  from  the  first  of  the  above  examples  that  there 
are  two  permutations  of  two  things.    Hence, 

To  find  the  permutations  of  three  letters,  «,  h,  c,  Ave  form 
Hiree  sets  of  permutations,  the  first  beginning  with  a,  the  sec- 
ond with  h,  and  the  third  with  c. 

In  each  set  the  first  letter  is  to  be  followed  by  all  possible 
permutations  of  the  remaining  letters,  namely : 

In  1st  set,    after    a    write    he,    ch,    making     ahc,    acb. 


ii 


a 


2d 
3d 


a 


a 


a 


a 


h 
c 


18 


ac,  ca, 
ah,  ha, 


a 


a 


har,     hca. 
cah,    cha. 


k> 


(1 


..  f 


?     I  ! 


J 


.)  :> 


A  'I 


''( 


'  / 


274 


PERMUTATIONS. 


I 


M 


Hence,  Pg  =  3-2  =  G. 

The  perm u tut  ions  of  n  thinfrs  can  l)e  divided  into  sels. 
The  tirst  set  befj^ins  with  tlie  first  thing,  followed  by  all  po.s.si- 
ble  permutations  of  the  remaining  n  —  1  things,  of  which  tln' 
number  is  Pn -i.  The  second  set  begins  with  the  second  thiuir, 
followed  by  all  possible  permutations  of  the  remaining  n  —  1 
things,  of  which  the  number  is  also  Pn-h  ^inJ  so  with  all  // 
sets.  Hence,  whatever  be  n,  there  will  be  n  sets  of  Pn-i  per- 
mutations in  each  set.     Therefore, 

Pn  =  nPn-\. 

This  equation  enables  us  to  find  Pn  whenever  we  know 
Pn-\i  and  thus  to  form  all  possible  values  (  f  P„,  as  follows: 


It  is  evident  that 
We  have  found 

Putting  ?z  =  4,  we  have 
n  =  5,   " 
etc. 


P,  =  1. 
Pi  =  2- 
P3  =  3.2-1  -31  =  6. 
4!  =  24. 


Pg  =  2-1  =  2! 


P,  =  4P3 


(( 


i( 


P,  =  5 

etc. 


P^  =  51  =  120. 
etc. 


It  is  evident  that  the  number  of  permutations  of  n  things 
is  equal  to  the  continued  product 

1.2.3.4 n, 

which  we  have  represented  bv  the  symbol  n !  so  that 


Pn  =  n ! 


1^ 


i , 


1% 


•      II 


EXERCISES.* 

1.  Write  all  the  permutations  of  the  following  letters  : 

bed,        acd,        abd,        abed. 

2.  What  proportion  of  the  possible  permutations  of  the 
letters  a,  e,  in,  t,  make  well-known  English  words? 

3.  Write  all  the  numbers  of  four  digits  each  of  which  can 
be  formed  by  permuting  the  four  digits  1,  2,  3,  4. 

4.  How  many  numbers  is  it  possible  to  form  by  permuting 
the  six  figures  1,  2,  3,  4,  5,  6. 

*  If  the  student  finds  any  difficulty  in  reasoning  out  these  exercises, 
he  is  recommended  to  try  similar  cases  i''  which  few  symbols  are  involved 
by  actually  forming  the  permutations,  until  he  clearly  sees  the  general 
principles  involved. 


PERMUTATIONS. 


5.  At  a  (linnor  party  a  row  of  G  pliilos  is  set  for  tlio  liost 
jind  T)  quests.  In  how  iiitiny  ways  may  they  he  seated,  siihjcot 
111  the  condition  that  tiie  hosl  nuist  iiave  Mr.  Brown  on  his 
rii::ht  and  Mr.  Hamilton  on  his  left  ? 

6.  Of  all  numbers  that  can  be  ftjrmed  by  permuting  the 
seven  digits,  1,  2  ....  7: 

((/)  How  many  will  be  even  and  how  many  odd  ? 
{!))  In  how  many  will  the  seven  digits  be  alternately  even 
and  odd  ? 

{c)  In  how  many  will  the  three  even  digits  all  be  together  ? 
(</)  In  how  many  will  the  four  odd  digits  all  be  together? 

7.  In  how  many  permutations  of  the  8  letters,  a,  b,  c,  d,  Cj 
f\g,h,wi\\  the  letters  d,e,f,  stand  together  in  alphabetical 
order  ? 

8.  In  how  many  of  the  al)0ve  permutations  will  the  Avord 
(Iccf  be  found  ? 

9.  In  how  many  of  the  permutations  of  the  first  0  letters 
^vill  the  words  ar/e  and  bid  be  both  found? 

10.  A  party  of  5  gentlemen  and  5  ladies  agree  v/ith  a  math- 
ciiKitician  to  dance  a  set  for  every  way  in  which  he  can  divide 
them  into  couples.     How  many  sets  can  he  make  them  dance? 

11.  In  how  many  of  the  permutations  of  the  letters  a,  b,  c, 
(/,  0,  \f'\\\  d  and  no  other  letter  be  found  between  a  and  c. 

12.  In  how  many  of  the  permutations  of  the  six  symbols, 
Hj  b,  c,  d,  e,f,  will  the  letters  dbo  be  found  together  in  (^no 
group,  and  the  letters  dcf'ni  another? 

13.  How  many  permutations  of  the  seven  symbols,  a,  b,  Cy 
(I  (',  f,  g,  are  possible,  subject  to  the  condition  that  some  per- 
mutation of  the  letters  ahc  must  come  first  ? 

14.  The  same  seven  symbols  being  taken,  how  many  per- 
mutations can  be  formed  in  which  the  letters  cd}c  shall  remain 


together  ? 


Perinii cations  of  Sets. 


353.  Def.  When  permutations  are  formed  of  only 
s  things  out  of  a  whole  number  7^,  they  are  called  Per- 
mutations of  it  things  taken  s  at  a  time. 


r* 


', 


r' 


;ti!i 


ip ' 


270 


rim  MUTATIONS. 


\  '    i 


Example.  'IMr'  permutations  of  thu  three  letters  a,  />,  r, 
taken  two  iit  a  time,  are 

ah,    ha,     ac,     ca,     he,     ch. 

The  i)ennutiitiuns  of  1,  !;J,  3,  4,  taken  two  at  a  time,  are 

1-v*,   i;i,  14,  ^1,  n,  'i\,  ;U,  '.VI,  34,  41,  4-.>,  43. 

Pkohlkm.  To  pud  the  miinbev  of  pcrinutatLoiis  oj' 
n  thiiiils  taken  s  at  a  time. 

Suppose,  first,  that  we  take  two  tilings  at  a  time,  as  in  the 
above  examples.  We  may  choose  any  one  of  tiie  n  things  as 
the  first  in  order.  Whiei)  one  soever  we  take,  we  shall  ha\i' 
n  —  1  left,  any  one  of  which  may  be  taken  as  the  second  in 
order.     Hence,  the  permutations  taken  2  at  a  time  will  be 

[Compare  with  the  last  example,  where  n  =  4.] 
To  form  the  i)ermutations  3  at  a  time,  we  add  to  each  per- 
mutation by  2's  any  one  of  the  n  —  2  things  which  are  lei'L. 
Hence,  the  number  of  permutations  3  things  at  a  time  is 

n  (n  —  1)  {71  —  2). 

In  general,  the  permutations  of  )i  things  taken  s  at  a  time 
will  be  equal  to  the  continued  product  of  the  s  factors, 

n  {71  —  1)  {n  —  2)  .  .  .  .  {n  —  s  -{-  1), 

n! 

which  is  equal  to  the  fiuotient    -. 

{n — s)\ 

It  will  be  remarked  that  when  s  =  7i,  we  shall  have  the 
case  already  considered  of  the  permutations  of  all  n  things. 

EXERCISES. 

1.  Write  all  the  numbers  of  two  figures  each  which  can  be 
formed  from  the  four  digits,  3,  5,  7,  9. 

2.  Write  all  the  numbers  of  three  figures,  beginning  with 
1,  which  can  be  formed  from  the  five  digits,  1,  2,  3,  4,  5. 

3.  How  many  different  numbers  of  four  figures  each  can 
be  formed  with  the  digits  1,  2,  3,  4,  5,  6,  no  figure  being  re- 
peated in  any  number  ? 


rEHMUTATlONti. 


277 


4.  Expliiin  'ow  all  the  mi!n))ers  in  tlio  preceding  exercise 
ijiiiy  be  written,  sliowing  how  many  niiinheiy  hi'^nn  witii  1, 
jiitw  many  with  'i^  etc. 

5.  In  lunv  nnmy  ways  can  .'^  gcnth'tnen  select  tlieir  partners 
froni  5  ladies? 

().  How  many  even  nunihers  of  3  diilerent  digits  each  can 
be  formed  from  the  seven  digits,   i,  'Z,  .  .  ,  .  I'i 

7.  How  many  of  these  nunihers  will  consist  of  an  odd 
diuit  between  two  even  ones  ? 


Circular  I'ermutatioiis. 

*25I{.  If  we  have  the  three  letters  a,  h,  r,  arranged  in  i\ 
circle,  as  in  the  adjoining  figure,  then, 
however  we  arrange  them,  we  shall  lind 
them  in  alphabetical  order  b>  beginning 
with  a  and  reading  them  in  the  suitable 
direction.  Hence,  there  are  only  two 
(lilferent  circular  arrangements  of  three 
letters  instead  of  six,  namely,  the  two 
directions  in  which  they  may  be  in  al- 
l)habetical  order. 

Next  suppose  any  number  of  symbols,  say  a,  b,  c,  d,  e,  f,  g, 
//,  and  let  there  be  an  equal  number  of  positions  around  the 
circle  in  which  they  may  be  placed.  These  positions  are  num- 
bered 1,  2,  3,  4,  5,  0,  7,  8. 

For  every  arrangement  of  the  sym- 
bols we  may  turn  them  round  in  a  body 
without  changing  the  arrangement. 
Eiich  symbol  will  then  pass  through  all 
eight  positions  in  succession. 

By  performing  this  operation  with 
every  arrangement,  we  shall  have  all 
possible  permutations  of  the  eight  things 
among  the  eight  positions,  the  number 
of  which  is  8!,  which  are  therefore  eight  times  as  many  as 
the  circular  arrangements. 


iv 


I    ■ 

I       1 


HI 


278 


PI'UIMUTATIONS 


8_! 


Hence  tlic  miml)or  of  diHereiit  circulur  arrangements  U 
,  wliicli  is  the  suiiie  as  7! 


rv 


In  <^('ru'riil,  if  we  represent  the  nuniher  of  circulur  urran<5f(> 
iiR'iits  of  n.  tilings  i)y  ('«,  we  siiull  lia\e 

C„  =  {n-\)\ 

The  same  result  nuiylje  reaelieU  hy  the  following  reasoniiiu. 
To  I'orni  a  circular  arrangement,  we  may  take  any  one  symhol, 
a  for  example,  put  it  into  a  ilxed  position,  say  (1),  and  leave  it 
there. 

All  possihlc  arrangements  of  the  symbols  will  then  he 
formed  by  permuting  the  remaining  symbols  among  the  w- 
nmiuing  positions.     Hence, 


as  before. 


iVi  =  {n-\)\ 


EXERCISES. 


1.  In  how  many  orders  can  a  party  of  7  persons  take  their 
places  at  a  round  table? 

2.  In  how  many  orders  can  a  host  and  7  guests  sit  at  a 
round  table  in  order  that  the  host  may  litive  the  guest  of  higii- 
est  rank  npon  his  riglit  and  the  next  in  rank  on  his  left? 

3.  Five  works  of  four  volumes  each  are  to  be  arranged  on 
a  circular  shelf.  How  many  arrangements  are  possible  whidi 
will  keep  the  volumes  of  each  set  together  and  in  proper  order, 
it  being  indifferent  in  which  direction  the  numbers  of  the 
volumes  read. 

4.  In  how  many  circular  arrangements  of  the  5  letters  0,  b, 
e,  (1,  p,  will  a  stand  between  h  and  d  ? 

5.  If  the  10  digits  are  to  be  arranged  in  a  circle,  in  how 
many  ways  can  it  be  done,  subject  to  the  condition  that  even 
and  odd  digits  must  alternate  ?     (Note  that  0  is  even.) 

6.  The  same  thing  being  supposed,  how  many  arrange- 
ment's are  possible,  subject  to  the  condition  that  the  even  digiiS 
must  be  all  together  ? 

7.  In  how  many  circular  arrangements  of  the  first  six  let- 
ters will  the  word  deaf  he  found?  What  will  be  the  difference 
of  the  results  if  you  are  allowed  to  spell  it  in  either  direction? 


J'h'JiMiTATJONS. 


270 


PeriP'itatioii.s  wlifii  Sc'vcrjil  of  the  Tliiii^^s  are 

hli'iidt'ul. 

!:i5-l,  li'  tlu!  suiiK!  thinj,^  iippcurs  scvcriil  tiiiios  iun()M<T  the 
lliin«(s  to  bo  inTimited,  (he  iiiimlicr  oi'  ililTcrcnt  i)eriuutatioiis 
will  bo  loMS  tliiiii  wlion  the  things  are  all  (lilleront. 

ExAMi'LE.     The  iiorinuhilions  of  (laM  lire 

((((bb,     alxih,     nhba,     baab,     buba,     bbaa,  (1) 

which  jiro  only  six  in  number. 

Pu()Hij:>r.  Tt>  /ufd  Ike  nitnihcr  of  pevviatidiuiLS  icIlcil 
scvei'fd  <)/'  the  thing's  (we  ulcntictd. 

Lot  us  first  examine  how  all  Ji-1  ))('rmutations  of  4  things 
may  i)0  formed  from  the  above  0  permutations  of  adbb.  TA't 
us  distinguish  the  two  a^s,  and  tlie  two  //s  by  accenting  one  of 
each.  Then,  from  each  permutation  as  written,  four  may  1)0 
formed  by  permuting  the  similar  letters  among  themselves. 
For  exam})le,  taking  abbtt^  and  writing  it  abb'a' ,  wo  have,  by 
permuting  the  similar  letters, 

ab'ba,     a'b'ba.     abb'a',     a'bh'a.  {'I) 

In  the  same  way  four  permutations,  differing  only  in  the 
{UTangement  of  the  accents,  may  be  formed  from  each  of  the 
G  j)ermutations  (1),  making  ;i-4  in  all,  as  there  ought  to  bo. 
(§  ;>51.) 

Generalizing  the  preceding  operation,  we  roach  the  iollow- 
iiig  solution  of  our  problem.  Let  the  symbols  to  be  permuted 
be  a,  b,  c,  etc. 

Suppose  that  a  is  repeated  r  times, 

i(  i(        h    '^  '<  V  '^ 

it  il         -    <<  it  f  ii 

etc.  etc.  etc. 

and  let  the  whole  number  of  symbols,  counting  repetitions,  bo 

11,  so  that 

?i  r=  r  +  -s'  +  /  +  etc. 

[Li  the  preceding  example  (1),  7i  =  4,  r  =  2,  .s  =  2.] 
Also  put  Xn,  the  required  number  of  different  permutations 
of  the  n  symbols. 


II  » 


n 


280 


I'EUMUTATJON^. 


!    i  I 


>; 


iii 


* 


Suppose  tliese  Xn  different  pornmttitions  iill  written  out, 
and  su})pose  tlie  symbols  wliicli  are  repeated  to  be  distingiii.shed 
])y  aeeents.     Tiieu : 

From  each  of  the  Xn  permutations  may  be  formed  P,-  =  r  I 
permutations  ))y  permuting  the  a^s  among  themselves,  as  in 
(2).     We  shall  then  have  r!X«,  permutations. 

From  each  of  the  latter  may  be  formed  .s!  permutations  l»v 
permuting  the  i's  among  themselves.  AVe  shall  then  have 
til  r\  X  X'n  permutations. 

From  each  of  these  may  be  found  f\  permutations  by  in- 
terchanging the  (fa  among  themselves. 

Proceeding  in  the  same  way,  we  shall  have 

Xn  X  rl  X  s\  X  f\  X  etc. 

possible  permutations  of  all  71  things.     But  this  number  has 
been  shown  to  be  n\     Therefore, 

Xn  X  r\  X  si  X  tl  X  etc.  =z  n I 


By  division, 


An  — 


n 


(•>) 


rl  si  tl  etc' 

which  is  the  required  ex])ression. 

We  remark  that  if  any  symbols  are  not  repeated,  the  for- 
mula (o)  will  still  be  true  by  supposing  the  number  corresi3on(l- 
ing  to  r,  s,  or  t  to  be  1. 

EXAMPLES. 

lo  The  number  of  possible  permutations  oi  aabh  are 

4'  24 

—,---:  =:  — — -  =  f),  as  already  found. 

2.  The  possible  permutations  of  aanhhcd  arc 

7 !  5040 


3!  2! 


0-2 


420. 


EXERCISES. 

Write  all  the  permutations  of  the  letters: 

I.     aadb.  2.     aahc.  3.     aaabc. 

4.  How  many  different  numbers  of  seven  digits  each  can 
be  formed  ))y  permuting  the  figures  1112225  ? 


rEUMUTA  T10N8, 


281 


3  number  lias 


5.  If  every  different  pernmtation  of  letters  made  a  word, 
liow  many  words  of  13  letters  eauli  couid  be  formed  from  the 
word  Masmch usetts. 

The  Two  Classes  of  Pennutations. 

255.  Tlie  n\  possible  permutation-'  of  h  tlii..gs  are  divisi 
ble  into  two  classes,  commonly  distin^^uisbed  as  even  permu- 
tations and  odd  })ernuitations  in  tlie  following  way: 

We  suppose  the  n  things  first  arranged  in  alphabetical  or 
munerical  order, 

a,  h,  c,  d,  .  .  .  .        or        1,  2,  3,  4,  ...  .  n, 

and  we  call  this  arrangement  an  even  pcrmntation. 

Then,  haying  any  other  permutation,  we  count  for  each 
thing  how  many  other  things  of  lower  order  come  after  it,  and 
take  the  sum. 

If  this  sum  is  even,  the  permutation  is  an  even  one ;  if  odd, 
an  odd  one. 

EXAMPLES. 

I.  Consider  the  permutatiim  265143. 


Here  2  is  followed  by  1  number  of  lower  order,  namely,  1. 
'•'    C     ''         "        4       ''        "  "  ''       5,1,4,3. 

ii    5     ''         "        3       "        "         ''  "       1,4,3. 

a     -\      '(         ^<         n       <<         >•  '< 

((    A.     "         ^'         1       *'         '*  "  **       3 

Then  1  +  4  + 3  4-0  +  1  =  9.     Hence  the  permutation  is  odd. 
?..  Consider  cdbea. 

Here  c  is  followed  by  2  letters  before  it  in  order,  namely,  ha. 
u    ^i    ,i         i:         2      "  "  "  "  "    ha. 


a 


a 


(( 


i( 


a 


a 


i( 


(( 


(( 


a 


(< 


a 


a. 


c     •'  '•  1       ••  *•  *•  •'        a. 

Then  2  +  2  +  1  +  1  =  0.     Hence  the  permutation  is  even. 

Def.  The  total  number  of  times  winch  a  thing  less 
m  order  follows  one  greater  in  order  is  called  the 
Number  of  Inversions  in  a  permutation. 


•  i 


I,     ! 


i  ;  i 


f 


1  ;i 


1. 


,.i 


282 


PERMUTATIONS. 


II 


Example.  In  the  preceding  permutation,  !^G5143,  the 
nuinl)er  of  inversions  is  'J.     In  cdbed  it  is  (J. 

JtEy.  It  will  bo  seen  that  the  ehiss  of  a  permntation  is 
oven  or  odd,  according  as  the  iiumljer  of  inversions  is  even  or 
odd. 

I'jiKOiiEM  I.  //,  iti  a  pcnnutdbloii,  t.'vo  tldiigs  arc, 
interchcDigcd,  the  class  will  he  chaiigcd  from  even  to  odd, 
or  from  odd  to  even. 

Proof.  C^)nsider  first  the  case  in  which  a  pair  of  adjoiiiiii'f 
things  are  interchanged.     Let  ns  call: 

ik,  the  two  things  interchanged. 

Ay  the  collection  of  things  which  precede  i  and  k. 

C,  the  collection  of  things  which  follow  them. 

The  first  permutation  will  then  be 

yia-6'.*  [a] 

After  interchanging  i  and  k,  it  will  be 

AkiO.  {!>) 

Because  the  order  of  things  in  A  remains  undisturbed,  each 
thing  in  A  is  followed  by  the  same  things  as  l^efore.  In  the 
same  way,  each  thing  in  C  is  preceded  by  the  same  things  as 
before. 

Hence,  the  number  of  times  tiiat  each  thing  in  A  or  C  is 
followed  by  a  thing  less  in  order  remains  unchanged,  ami, 
leaving  out  the  pair  of  things,  /,  k,  the  number  of  inversions 
is  unchanged. 

But,  by  interchanging  i  and  k,  the  new  inversion  kl  is  in- 
troduced. Therefore  the  numb','r  of  inversions  is  increased 
by  1. 


■•^'  This  fdrrti  of  algebraic  notation  differs  from  those  already  used  in 
that  the  .symbols  A  and  C  do  not  stand  for  quantities,  but  mere  coll;*- 
tions  of  letters.  It  is  an  application  of  the  general  princijilf^  that  a  sitigli' 
symljol  may  be  used  to  represent  any  set  of  synd)ols,  but  must  represent 
the  same  set  throughout  the  same  (luestion.  A  and  C  are  hero  used  to 
show  to  the  eye  that  in  forming  the  permutations  of  (6)  from  {a),  all  the 
h>ttera  ou  each  side  of  ilc  preserve  their  relative  positions  unchanged. 


PERMUTATIONS. 


283 


If  the  first  arrangement  is  ki,  this  one  inversion  is  removed. 
Ilcnce,  in  either  case  the  number  of  inversions  is  changed  by 
1.  and  is  therefore  changed  from  odd  to  even,  or  vice  versa. 

lllastration.     In  the  permutation 

205143, 

the  inversions,  as  ah-eady  found,  are  the  following  nine  : 

21,     05,     01,     0-i,     03,     51,     54,     53,     43. 

Let  us  now  interchange  5  and  1,  making  the  [)ermutation 

201543. 

The  inversions  now  are 

21,     01,     05,     04,     03,     54,     53,     43, 

the  same  as  before,  except  that  51  has  been  removed. 

Next  consider  the  case  in  which  the  things  interchanged 
do  not  adjoin  each  other.     Suppose  that  in  the  permutation 

b  a  d  e  h  c  f 

wo  interchange  a  and  h.  We  may  do  this  l)y  successively  in- 
terchanging a  with  d,  with  e,  and  with  h,  making  three  inter- 
changes, producing 

h  d  e  h  a  c  f . 

Then  we  interchange  h  with  e  and  with  d,  making  two 
interchanges,  and  producing 

b  h  d  G  a  c  f  y 

which  effects  the  required  interchange  of  a  with  h. 

Tlie  number  of  the  neighboring  interchanges  is  3-f  2  =  5, 
an  odd  number.  Because  the  number  of  inversions  is  changed 
from  odd  to  even  this  same  odd  number  of  times,  it  will  end 
in  the  opposite  chiss  with  which  it  commenced. 

Theorem  II.  TJic  possible  permutations  of  n  tilings 
are  one-half  even  and  one-linlf  odd. 

Proof.    Write  the  n !  possible  permutations  of  tlie  7i  things. 

Then  interchange  someone  pair  of  things  {e.g.,  the  first 
two  things)  in  each  permutation.  We  shall  have  the  same 
})ermutations  as  before,  only  differently  arranged. 


I  »' 


I-.! 


IM 


1 


..'    ,  .1 


•i 

i-i 


284 


PERMUTATIONS. 


() 


f: 


k 

1  i 


?'fe 


r. 


By  tlic  change,  every  even  permutation  will  be  changed  \ 
odd,  and  every  odd  one  to  even. 

Because  every  odd  one  thus  corresponds  to  an  even  ono. 
and  vice  versa,  tlieir  numbers  must  be  e(jual. 

Illustration.     The  permutations  in  the  second  column  I'nl 
lowing  jire  formed  from  those  in  tlie  first  by  interchanging  the 
first  two  figures  : 

1  2  8        even,  2  1  3        odd. 
13  2         odd,  3  12         even. 

2  13        odd,  12  3        even. 

2  3  1        even,  3  2  1        odd. 

3  12        even,  13  2        odd. 
3  2  1         odd,                      2  3  1        even. 

EXERCISES. 

Count  the  number  of  inversions  in  each  of  the  foUowinir 
permutations: 

I.     hcdngef.  2.     hcagdef.  3.     325941. 

4.     5432.  5.     829173G4.  6.     82971364. 

3^0  Def.  A  Symmetric  Function  is  one  which  is 
not  changed  by  pei'iiiuting  the  symbols  which  enter  into  it. 

An  Alternating  Function  is  one  which,  wdien  any  two 
of  its  symbols  are  interchanged,  changes  its  sign  without 
changing  its  absolute  value. 

EXERCISES 

Show  which  of  the  following  functions  are  symmetric  and 
which  are  alternating : 

I.  a  -\-  h  -\-  c.  2.     ahc. 

3.  a{h  +  c)  ^-  h{c  +  a)  +  c{a  +  h). 

4.  a"  (b-c)  +  h'  {c-a)  +  c'  (a  -  b). 

5.  a'  {b  +  c)  +  ¥  {c  +  a)    1  c'  {a  -f  h). 

6.  (a—b)  (6  —  c)  {c  —  a). 

7.  ab  -{-  be  +  ca. 


COMBINA  TIONS. 


285 


even  one. 


metric  ami 


CHAPTER     II. 

COMBINATIONS. 

35T.  Def.  The  number  of  ways  in  which  it  is  pos- 
sible to  select  a  set  of  s  things  out  of  a  collection  of  it 
things  is  called  tlu.*  Number  of  Combinations  of  s 
things  in  n. 

Ex.  I.  From  the  three  synihols  a,  h,  c,  may  he  formed  the 

couplets, 

ah,        ac,        he. 

Ilciice  there  are  three  combinations  of  2  things  in  o. 

Ex.  2.  P^rom  a  stud  of  four  horses  may  be  f(jrmed  six  dif- 
ferent span.     If  we  call  the  horses  A,  B,  C,  D,  the  different 

s})un  will  be 

AB,     AC,     AD,     BC,     BD,     CD. 

Rem.  1.  A  set  is  regarded  as  different  when  any  one  of  its 
separate  things  is  different. 

Rem.  2.  Combinations  differ  from  permutations  in  that, 
in  forming  a  combination,  no  account  is  taken  of  the  order  of 
arrangement  of  things  in  a  set.  For  instance,  ah  and  ha  are 
tlie  same  combination.  Hence,  we  may  always  suppose  the 
letters  or  numbers  of  a  coml)i nation  to  be  Avritten  in  alpha- 
betical or  numerical  order. 

Notation.  The  number  of  combinations  of  s  things  in  n 
is  sometimes  designated  by  the  symbol, 

Problem.  To  find  the  niunher  of  combi nations  of  s 
things  in  n. 

If  we  form  ever)'  possible  set  of  .<?  things  out  of  n  things, 
and  then  permute  the  s  things  of  each  set  in  every  possible 
way,  we  shall  have  all  the  permutations  of  n  things  taken  s  at 
a  time  (§  252).     That  is, 


;      I 


.r  .1 


!  t 


^■r 


fi; 


.,. 


'  I 


f» 


il 


28G 


COMBINATIONS. 


I 


!'• 


m. 


;iH> 


:i» 


exprt'ss  (lie  number  of  pL'rnnitatioiis  of  u  tilings  taken  s  at  a 
time.     But  we  luive  found  this  number  to  be 

n{)i  —  i)  {n  —  'Z)  .  .  .  .  (?^  —  s  +  1). 

We  liave  also  found 

1'   —  .s'!  r=  1.2.3.4 .9. 


Ileuec,     Cg  X  6- !  =^  n  {11  —  1)  {n  —  'l)....{n—-s-\-  1), 

..nrl  r'«  -  ''  ('^  -  1)  («  -  2) {n-s  +  1) 

l-^-J.-i  .  .  .  .  a; 


=  (^]  (i5  228,  3)  ; 


or 


n\ 


s\  {n-s)y 


Avliicli  is  the  reqnired  expression. 

Rem.  For  every  combination  of  s  things  which  wo 
can  take  from  n  things,  a  combination  oi  n—  s  things 
will  be  left. 

Hence,  C^  =  Cls. 

This  formula?  may  be  readily  derived  from  the  expression 
for  the  number  of  combinations.    For,  if  we  take  the  equation 

this  formula  remains  unaltered  Avhen  we  substitute  n  —  s  f(jr 
s,  and  therefore  also  represents  the  combinations  of  n  —  ,v 
things  in  n. 

Def.  Two  combinations  which  together  contain  all 
the  things  to  be  combined  are  called  two  Complement- 
ary combinations. 

EXERCISES. 

1.  Write  all  combinations  of  two  symbols  in  the  five  sym- 
bols, a,  b,  c,  il,  e. 

2.  Write  all  combinations  of  three  symbols  in  the  same 
letters,  and  show  why  the  number  is  the  same  as  in  Ex.  i. 


COMBINATIONS. 


287 


3.  A  spun  of  horses  being  dilTeretit  when  cither  liorsc  is 
('hanged,  how  many  different  span  may  be  formed  from  a  stud 
of  3?     Of  7?     Of  9? 

4.  If  four  points  are  marked  on  a  piece  ol  paper,  how  many 
distinct  lines  can  be  formed  l)y  joining  tliem,  two  and  two? 
Itow  many  in  the  case  of  ?^  points? 

From  each  one  of  the  points  can  ))c  drawn  n  —  I  lines  to 
other  points;  then  why  are  tiiere  not  n{n  —  l)  lines? 

5.  If  iive  lines,  no  two  of  which  are  parallel,  intersect  each 
other,  how  many  points  of  intersection  will  there  be?  llow 
many  in  the  case  of  n  lines  ? 

6.  If  n  straight  lines  all  intersect  each  other,  how  many 
(litferent  triangles  can  be  found  in  the  figure? 

7.  In  how  many  different  ways  may  a  set  of  four  things  be 
tlivided  into  two  pairs  ? 

8.  In  how  many  ways  can  a  party  of  four  form  partners  at 
whist  ? 

9.  In  how  many  ways  can  the  following  numbers  be  thrown 
with  three  dice : 


(a)     1,  1,  1 ; 


{h)     1,2,2; 


{c)     1,2,3. 


10.  A  school  of  15  yoi  ng  ladies  have  the  privilege  of  send- 
ing a  party  of  5  every  day  to  a  picture  gallery,  provided  they 
do  not  send  the  same  party  twice.  How  many  visits  can  they 
make  ? 

Combi^iations  with  Repetition. 

358.  Sometimes  combinations  are  formed  with  the  liberty 
to  repeat  the  same  symbol  as  often  as  wo  please  in  any  set. 

Example.  From  the  three  things  a,  h,  c,  are  formed  the 
six  combinations  of  two  things  with  repetition, 

ab,        ac,        hi,        he, 


aa. 


cc. 


Problem.     To  find  the  numher  of  combinations  of  s 
tilings  in  n,  luhen  repetition  is  allowed. 

Solution.     Let  the  n  things  be  the  first  n  numbers, 

1,    i«,    O,    4;,    .    .    •    .    11. 


\v 


:i>4 


11 


■j'!^ 


.,»    1 


288 


COMBTNA  TIONS. 


•p.,: 


Form  all  ])ossi})le  sets  of  s  of  these  numbers  with  repetition, 
the  numl)ers  of  each  set  being  arranged  in  nunierieal  order. 

Let  lis  ho  the  re(|uired  number  of  sets.     Then,  in  each  set, 

Let  the  first  numl)er  stand  unchanged. 
Increase  the  :^d  number  by  1. 

"        "    ;jd       *'         "2. 

"        "    4th     ''        "  3. 


t( 


(( 


i-th 


a 


8  — 


|i>' 


•    t 


j 


:,  "% 


!*• 


"We  shall  then  have  Eg  sets  of  s  numbers,  each  without  rep- 
etition. 

Example.    From  the  numbers  1,  2,  3  are  formed  with  repetition, 

11,     12,    13,    23,    23,    33. 
Then,  increasing  the  second  numbers  by  1,  we  have 

13,    13,    14,    23,    24,    34. 

The  greatest  possible  number  in  any  set  after  the  increase 
will  be  n  -\-  s  —  1,  because  the  greatest  number  from  whieii 
the  selection  is  made  is  n,  and  the  greatest  quantity  added  i.s 
s  —  1.  Hence  all  the  new  sets  will  consist  of  combinations  of 
s  numbers  each  from  the  n  -\-  s  —  1  numbers, 

1,  2,  3,  4,  .  .  .  .  ?i  .  .  .  .  7i  +  s  —  1.  (n) 

No  two  of  these  combinations  can  be  the  same,  because  then 
two  of  the  original  combinations  would  have  to  be  the  same. 
Hence  the  new  sots  are  all  diiferent  combinations  of  s  numbers 
from  the  n -\- s — 1  numbers  {a).  Therefore  the  number  of 
combinations  cannot  exceed  the  quantity  C^. 

Conversely,  if  we  take  all  possible  combinations  of  s  differ- 
ent numbers  in  n  -\-  s  —  1,  arrange  each  in  numerical  order, 
and  subtract  1  from  the  second,  2  from  the  third,  etc.,  we 
shall  have  different  combinations  from  the  first  n  numbers 
with  repetitions.  Hence  the  number  of  combinations  in  the 
second  class  cannot  exceed  those  of  the  first  class. 

Hence  we  conclude  that  the  number  of  combinations  of  s 
things  in  n  with  repetition  is  the  same  as  the  combinations  of 
s  things  in  n  -[-  s  —  1  without  repetition,  or 


COMBINAriONS. 


289 


7?;=  C"-^  =  C--7~   ) 

)t  {n  +  1)  {n  ^-  ^^)  .  .  .  .  {n  -f-  .s 


1) 


i--^.;3.4 


EXERCISES. 


.   .   tS 


1.  Write  all  possible  conibiiiatioiis  of  o  numbers  with  repe- 
tition out  of  the  three  numbers  1,  '1, 3  ;  then  inerease  the  seet)n(l 
of  eaeli  combination  by  1  and  the  third  by  2,  and  show  that 
we  have  all  the  combinations  of  three  different  numbers  out  of 
1,  2,  3,  4,  5. 

2.  IIow  many  combinations  of  4  things  in  4  with  repeti- 
tion?    Of  H  things  in  u  ? 

In  the  last  question  and  in  tin*  following,  reduce  the  result  to  itH 
lowest  ternas. 

3.  How  many  combinations  of  n-{-l  things  in  w  —  1  with 
repetition  ? 

Special  Cases  of  C/Oiubiiiatioiis. 

359.  It  is  plain  that 

cr  =  n, 

l)ccause  eacli  of  these  combinations  consist  simply  of  one  of  the 
n  things.     Hence,  also, 

Cl-i  =  n, 

because  in  every  such  combination  one  letter  is  omitted. 
It  is  also  plain  that 

because  the  only  combination  of  71  letters  is  that  comprising 
the  n  letters  themselves.    Hence  we  write,  by  analogy, 

C^o  -  1, 

although  a  combination  of  nothing  does  not  fall  within  the 
original  definition  of  a  combination. 

360.  The  formulee  of  combinations  sometimes  enable  us 
to  discover  curious  relations  of  numbers. 

1.  Let  us  inquire  how  we  may  form  the  combinations  of 


1 1  if 


»,.•:; 


■'•:i 


I .'  I 


290 


COMB  IN  A  TI0N8. 


w 


K- 


'WH^ 


1  '^M 


5+1  tilings  when  we  liave  those  of  s   tilings.    Let  the  n 

things  from  which  the  combinations  are  to  be  formed  be  the 

letters 

a,  b,  c,  (I,  pyf,  (J,  etc (n  in  number). 

Lot  all  the  combinations  of  .s+  1  of  these  n  letters  bo  writ- 
ten in  alphabetical  order.     Then: 

1.  In  the  combinations  beginning  with  r/,  the  letter  a  wi'l 
be  followed  by  all  possible  combinations  of  .s  letters  out  of  tin.* 
n  —  \  letters  b,  c,  d,  etc.,  of  wliich  the  number  is  f'a  ^' 

)i.  In  the  combinations  beginning  with  b,  tiic  letter  b  is 
followed  by  all  combinations  of  s  letters  out  of  the  n  —  2  let- 
ters c,  d,  e,  f,  etc.  Therefore  there  are  6'"'^  combinations 
beginning  with  b. 

3.  In  the  same  way  it  maybe  shown  that  there  are  6'"  '^ 
combinations  beginning  with  c,  CT  beginning  with  d^  etc. 
The  series  will  terminate  with  a  single  combination  of  the  last 
s  +  1  letters. 

Since  we  thus  have  all  combinations  of  6'  +  l  letters,  W(! 
find,  by  summing  up  those  beginning  with  the  several  letters 
a,  i,  c,  etc., 

pn-\         pn-2         ^n-3  i     r^  —   r'>^  /„\ 

Substituting  for  the  combinations  their  values,  we  find 

'n  —  21 

I  +  I f  -4-  ....   4-  I  -  I   =  I 


i^)  ^  c^) + c^v  •  •  •  -  (i) = m- 


By  the  notation  (§  228,  3),  all  the  terms  of  the  first  member 
have  the  common  denominator  s\,  while  the  numerators  arc 
each  composed  of  the  factors  of  s  consecutive  numbers.  Mul- 
tiplying both  sides  by  si  and  reversing  the  order  of  terms  in 
the  first  member,  we  have 

1.2-3 s  +  2.3-4 s  +  1  +  etc.  1 

etc.  etc. 

+  {n  —  s  —  1)  .  .  .  .  {71  —  3)(n  —  2) 
+  (n  —  s)  .  .  .  .  {n  —  2){n  —  1) 

_  (n  —  s)  .  .  .  .  (n  —  2)  {n  —  l)n 


COMlilNA  TIONS. 


201 


The  student  Ib  now  recomiuonth'd  to  go  ovor  the  prccodinfj:  process 
with  spcriul  simple  nuinericul  vultu's  of  n  and  h  whicli  ho  may  seh^ct  for 
himself. 

EX.AMPLES. 

If  n  =z6  and  .s  =  "Z,  \vc  have 

l.o  ^  0.3  ^_  3.4  _ 

If  7i  —  7  unci  5  =  3. 

1-^.3  +  2-3.4  -f  3.4.5  4-  -l-S.G  = 
If  n  =  7  and  s  =  4^, 

1.3. 3. •4  +  2.3. 4.5  +  3. 4.5.0  = 
If  n  =  0  and  s  =  3, 


3.4.5 


4.5.fi.7 


3.4.5.0.7 


5 


1.2.3  +  ^.3.4  +  3.4.5  +  4.5.0  +  5.0.7  +  0.7.8  = 


0.7.8.9 


Prove  these  efjiiations  by  computing  both  members. 

SOI.  Another  cnrious  example  is  the  following: 

Let  lis  have  p  +  q  things  divided  into  two  sets,  the  one 
containing  p  and  the  otlier  q  things.  Then,  to  form  all  possi- 
ble combinations  of  s  things  out  of  the  wliole  2>  +  (/>  ^^'^  n^^iy 
take : 

Any  s  things  in  set  7; ; 

Or  any  combination  of  s  —  1  things  in  set^;  with  any  one 
tiling  of  set  q ; 

Or  any  combination  of  5  —  2  things  in  set^;  with  any  com- 
bination of  2  things  m  q ', 

Or  any  combination  of  5  —  3  things  in  j)  with  any  3  out 
of  y,  etc. 

We  shall  at  length  come  to  the  combinations  of  all  s  things 
out  of  q  alone.  Adding  up  these  separate  classes,  we  shall 
have : 

C^s  +  cu  c\  4-  CU  CI  +  ....  +  C?  CU  +  CI 

This  sum  makes  up  all  combinations  of  s  things  in  the 
whole  p-\-q,  and  is  therefore  equal  to  C^s^^.  Putting  the 
numerical  expressions  for  the  combinations,  we  have  the 
theorem : 


I  !;'■' 


■'11 


t' 


,.i  I 


lifll 


>  :    ill 
ni    |i| 


ii.; 


202 


atMiiiXA  rioNs. 


I  • 


■•*i 


f-!,- 


\¥ 


If,  as  an  oxamplo,  wo  put  s  =  3,  j>  =  4,  q  =  5,  tliis  tlico 
rem  will  i>ivu 


n' 

!).S.7 


1.^.3  "^  l-a'  I  ■*■  TT."^  "^  1.5JV3' 


the  coiTOctucaH  of  which  is  caHily  proved  hy  computation. 

EXERCISES. 

1.  Write  all  the  conihiuations  of  three  letters  out  of  the 
five,  a,  h,  c,  d,  c,  juul  show  tiiat  i'\  of  tlieiu  Ix'^iu  with  ^/,  {"\ 
with  h^  and  C\  with  r,  according  to  the  reasoning  ol'  g  :i(j(). 

2.  Prove  that  C\  -  C\  +  C\, 

fifi  —  en    I    f^a 

m  gonoral,  (  g      =  (\  -{-  (  n-i- 

In  the  "ollowing  two  ways : 

(1.)  Let  all  comhinations  of  .s  letters  in  the  7i  letters 

a,  b,  c, ....  71 

be  formed,  tlioir  numl)or  being  (.^g.  Then  suppose  one  letter 
added,  making  the  nnml)er  n  +  1.  The  combinations  of  .s 
letters  out  of  these  71  +  1  Avill  include  the  Cg  formed  from 
the  71  letters,  plus  each  combination  of  the  additional  (^z  +  l)"^ 
letter  with  the  combinations  of  .s  —  1  out  of  the  first  7i  letlei's. 

(2.)  Prove  the  same  general  result  from  the  formula. 


c: 


(:)• 


3.  If  we  form  all  coml)inations  of  3  things  out  of  7,  how 
many  of  these  combinations  Avill  contain  a  7,  and  how  many 
will  not  ? 

4.  If  we  form  all  the  combinations  of  s  letters  out  of  the  n 

letters 

a,  h,  c,  .  .  .  .  ft, 


<U)MliL\A  TKlNS. 


2i):J 


how  many  of  tlioseconibinatioiis  will  contain  ^/,  and  how  many 
will  not  ? 

5.  In  tlio  ])n'C't'(lin^  case,  how  many  of  the  combinations 
will  contain  the  three  letters  a,  b,  c^ 

*HVl.  Tirr.oiiKM  T.  77ir  fohtJ  nifmhrr  of  r>)ni/u'nafionn 
irliich  cm/  he  j'oriiicd  from  n  thiiujs,  InrhuliiitJ  1  zero 
roinhiiKition,  is  :i". 

Jn  the  lan.<;na^^t'  of  Aljxi'hra, 

'  0  +  '  1  4"  '  'i  +  •  •  .  .  +  f'  rt  1  +  '  rt  =  *  • 

Proof.  Let  us  be<rin  with  'A  thin<,'s,  n.  h,  c,  and  let  us  call 
the  formal  zero  combinaliou,  1  =  Cq.     Then  we  have 

blank,  Number  =.  1 

a,  b,  c,  "        =  3 

ab,  ac,  be,  "        =:  :j 

abr,  «        =  ]_ 

Sutn  =::  S  =  '2^ 

Now  introduce  a  fourth  letter  d.  The  combinations  out  ot* 
the  four  things,  a,  b,  c,  d,  will  consist  of  the  above  8,  plus  the 
S  additional  ones  formed  by  writing,'  d  after  each  of  the  above 
eifjjht.     Their  number  will  therefore  be  10. 

In  the  same  way,  it  may  be  shown  that  we  double  the  pos- 
sible number  of  combinations  for  every  thinf^  we  add  to  the 
set  from  which  they  are  taken.     We  have  found,  for 

n  =  3,  Sum  of  combinations  =      8  =  2^; 

n  =  4,  «                 "              =  2-8  =  2<; 

n  =  5,  «                  "              =  2'2*=  2^; 

etc.  etc. 

which  shows  the  theorem  to  be  general. 

Theorem  II.  //  tJic  signs  of  fJir  nUernate  comhinrr- 
tions  of  n  things  ho  changed,  the  algebraic  sum  will  he 

zero. 


^3 

6"8, 


In  algebraic  language, 

Cl  -  Cl  +  Cl  -  C'l  +  etc.  ±Cl--^0. 


{a) 


t     I 


I  • 


!  '\ 


f] 


If 


'f. 


'  T? 


i  * 


^ 


294 


COMBINATIONS. 


Proof.    If  in  tlie  formula  of  §  1^01,  Ex.  2,  namely, 
wc  put  n  —  1  for  n^  it  becomes 

Putting  6'  successively  equal  to  0,  1,  2,  ...  .  n,  wc  have 

fin  f,n  -,   , 


'iU-l 


cir  +  ( 1 

fU-l 


n-1 


1  +  cV 


I* 


O  n-1  —    L/  ,t_2  -r  ^  n-l  —    '^  ?J-8  i"  -!■• 

Substituting  these  values  in  the  expression  (a),  it  becomes 

1  -  (i  +  cr')  +  (6'r  +  cr')  -  {cv  +  cr')  +  .... 

::^  1  _  1  _  or'  +  CT'  +  CV  -  CV  -  CV  +  etc. 

How  far  soever  we  carry  tliis  process,  all  the  terms  caned 
each  other  except  the  last.  Therefore,  if  we  continue  the  addi- 
tions and  subtractions  until  we  come  to  Cn-i ,  the  sum  will  bo 

C'l  -  C\  +  C\  -  etc ±  Cl-^  =r  ±  cVx  =  ±  1. 

The  last  term  will  be  =F  C^  —  T  1,  and  will  thercfuro 
just  cancel  the  sum  of  the  preceding  terms. 

Note.  Theorem  I  may  be  demonstrated  by  these  same  fonmiljp, 
since  the  sum  of  all  the  teruis  taken  positively  will  be  duplicated  every 
time  we  increase  n  by  1. 

30.3.  Independent  Combinations.  There  is  a  system  ol 
combinations  formed  in  the  following  wav  : 

lb  is  required  to  furni  a  emnhi nation  of  s  things,  hij 
taking  one  oitt  of  each  of  s  different  collections,  lloic 
incinij  coinhinations  can  he  formed  ? 

Let  the  1st  collection  contain  a  things, 

"       2d  "  ''       h      " 

a      3d  "  "       c      " 

etc.  etc. 


GOMBINA  TIONS. 


21)5 


Then  we  may  take  any  one  of  a  things  from  the  first  col- 
lection. 

AV^ith  each  of  tliese  we  may  combine  any  one  of  the  h  things 
ill  tlie  second  collection. 

With  each  of  these  we  may  combine  any  one  of  the  c  things 
of  the  third  collection. 

Continuing  the  reasoning,  we  see  that  the  total  number  of 
combinations  is  the  continued  product 

abc  ....  to  5  factors. 

If  the  number  in  each  collection  is  equal,  and  we  call  it  «, 
the  number  of  combinations  will  be  a*. 

This  form  of  combinations  is  that  which  corresponds  most 
nearly  to  the  events  of  life,  and  is  applicable  to  many  questions 
concerning  probabilities.  For  example,  if  any  one  of  five  dif- 
ferent events  might  occur  to  a  person  every  day,  the  number 
of  different  ways  in  which  his  history  during  a  year  might  turn 
out  is  5^^,  a  number  so  enormous  that  Ji5a  digits  would  be  re- 
(|uired  to  express  it. 

EXERCISES. 

1.  A  man  driving  a  span  of  horses  can  choose  one  from  a 
stud  of  10  horses,  and  the  other  from  a  stud  of  12.  Ilow 
iiumy  different  span  can  he  form  ? 

2.  It  is  said  that  in  a  general  examination  of  the  public 
schools  of  a  county,  the  pui)ils  spelt  the  word  scholar  iu  a530 
different  ways.     If  in  spelling  they  might  replace 

ch  by  c  or  k', 

0  by  an,  aw,  or  oo; 

1  by  U; 

a  by  e,  o,  ii,  or  ou',  ' 

r  by  re  ; 
in  how  many  different  ways  might  the  word  be  spelt  ? 

3.  If  a  coin  is  thrown  n  times  in  succession,  in  how  many 
different  ways  may  the  throws  turn  out  ? 

4.  If  there  are  three  routes  between  each  successive  two  of 
the  five  cities,  Boston,  New  York,  Philadelphia,  Baltimore, 
Wasliington,  by  how  many  routes  could  we  travel  from.  Boston 
lu  Washington? 


I '' 


I  I 


•I  I 


M  1 


296 


COMBINATIONS. 


.  ■  ♦ 


,1  i 


h 


I . 


Xi 


The  Binomial  Theorem  when  the  Power  is  a 

Wlioki  Number. 

204.  ''-I 'I  binomial  tlieorem  (S  IT-),  when  the  power  i.>^  a 
positive  integer,  can  be  demonstrated  by  the  doctrine  of  com- 
binations, as  follows: 

Let  it  first  he  required  to  form  the  product  of  the  n 
hiiiuDiial  factors, 

To  understand  the  form  of  the  product,  let  us  first  study  the  special 
case  when  w  =  3.  Performing  the  muhiplication  of  the  tirsst  three  fac- 
tors, the  product  will  'consist  of  eight  terms : 


This  product  is  the  expression  (a)  developed  when  n  —  Z. 


00 


We  conclude,  by  induction,  that  the  entire  product  (</) 
when  developed  in  this  same  way,  will  be  composed  of  a  sum 
of  terms,  each  term  being  a  product  of  several  literal  factors. 

When  (a)  is  thus  multiplied  out,  we  shall  call  the  result 
the  developed  expression. 

The  developed  expression  has  the  following  properties  : 

I.  Each  ^erin  coutaius  n  literal  factors,  as  and  x's, 
and  no  more. 

For,  suppose  x^^a^,  x^=ia^,  to  Xn  =  ((n.  Then  the 
expression  {a)  will  reduce  to 

and  tiKj  developed  expression  must  assume  lue  same  value : 
that  is,  it  must  consist  of  terms  each  of  which  reduces  to  the 
jxprci^sion 

when  we  change  x  into  a.  Noav  if  it  contained  any  term  with 
either  more  or  less  than  71  factors,  it  could  not  assume  this 
form. 


77/ A'  BINOMIAL    TIIEOUFM. 


297 


vver  IS  a 


II.  TJie  factors  of  each  tani  have  all  the  n  indices 

1     '^    '\  » 

For,  the  index  figure  of  no  term  is  iiltercd  by  eh;ni<ring  x 
into  a,  as  in  I.  Hence,  if  in  jiny  term  tiny  index  ligure  were 
iiiissinf,'  or  re])cated.  th;d  term  would  not  reduce  to  tlie  form 
( ■),  whence  tiierc  can  Ijo  neither  omission  nor  repetition  of 
;iny  index,, 

III.  Because  each  term  has  n  factors,  it  must  either 
liare 

n  factors  a; 

n  —  1  factors  a  and  one  factor  x ; 
n  —  2  factors  a  and  two  factors  x; 
In  general,  a  term  may  have  the  factor  a  repealed 
n  —  i  times,  and  x  repeated  i  times. 

IV.  In  a  term  winch  contains  i  factors  x,  tlieso  i  factors 
must  be  affected  with  some  combination  of  /  indices  out  of  tlie 
wliolo  number  1,  2,  3,  .  .  .  .  ?i ;  and  the  n  —  i  r/'s  must  be 
uifected  by  the  complementary  combination  of  n  —  i  indices. 
We  next  inquire  whether  there  is  a  term  corresponding  to 
every  such  combination.     Let 

be  any  combination  of  i  indices,  and 

/v,     0,    0,    o,   .   .   .   . 

the  complementary  combination  of  n  —  i  indices. 

Since  the  developed  expression  must  be  true  for  all  values 
of  a  and  X,  let  us  put  in  {a), 


rtj  ■—  0, 

«4  =  0, 

a^  ~  0, 
etc. 


0; 
0; 

0; 
0; 


00 


etc. 


The  product  {a)  will  then  reduce  to  the  single  term, 

x^a„x^x^a^a^x^a^  (0 

By  tlie  same  change  the  developed  expression  must  reduce 
to  this  same  value,  and  it  cannot  do  this  unless  the  expression 
{o)  is  one  of  its  terms. 


I'  I 


t 


\-\ 


H  H 


298 


COMBINATIONS. 


Vi" 


,    % 


Hence  the  developed  expression  imist  contain  a  term, 
corresponding  to  every  combination. 

V.  Since  every  oombinution  of  i  fif^ures  out  of  1,  2,  3, ....  /^ 
will,  ill  this  way,  give  rise  to  a  term  like  (i^;),  containing  llu' 
symbol  a  i  times,  and  the  symbol  x  n  —  i  times,  there  will  be 
6'f  such  terms. 


Now  suppose      «j  ~  f/g  =  «3  =  .  .  .  .  ctn  =  a. 

The  expression  {a)  will  then  reduce  to  {a  +  xy. 

In  the  developed  expression,  all  the  6'f  terms  containing  x 
i  times  and  a  n  —  i  times  will  now  be  equal  and  their  sum 
will  reduce  to  C'l  ^~' x\ 

Hence,  putting  in  succession  i  =  0,  i  =  1,  etc.,  to  i  =  n, 
we  shall  have 

(a+xY  =  a^-\-  Cia^-^xi-  Cla^-^x^  -\- -l-Cl  tax^~'^-{-x^ 

Substituting  for  Ci  its  value,  we  shall  have 

{a  +  x)»  ■—a^-{- 7ia^-^x  +  y  jr^"- V  +  ....  +  /— -.  jax''-^  +  (' j-'", 

which  is  the  Binomiid  Theorem,  enunciated,  but  not  demon- 
strated, in  Book  V,  Chapter  I. 

Note.  If  the  student  has  any  difficulty  in  understanding  the  ste])s 
of  the  preceding  demonstration,  he  should  suppose  n  —  3,  and  refer  tlio 
demonstration  to  the  developed  expression  (it'). 


«$  " 


-#■■•' 


PROBABILITIES. 


21)0 


CHAPTER    III. 

THEORY     OF     PROBABILITIES. 

205.  Def.  The  Theory  of  Probabilities  treats  of 
the  chances  of  tlie  occUxTence  of  events  which  cannot  bo 
foreseen  with  certainty. 

Notafion.  Let  a  l)ag  eonhini  \  bulls,  of  which  1  is  white 
and  3  black.  If  a  ball  be  drawn  at  raiidoiii  from  the  bag,  we 
!:liould,  in  ordinary  language,  say  that  the  chances  were  1  to  3 
in  favor  of  the  bah  being  white,  or  3  to  L  in  favor  of  its  being 
black. 

In  the  language  of  probabilities  wo  say  that  the  probability 

1  3 

of  a  white  ball  is  -,  and  that  of  a  black  one   ,  • 
4  4 

In  general,  if  there  are  m  chances  in  favor  of  an  event,  and 


n  chances  against  it,  its  probability  is 


m 


—  •     Hence, 


m  -\-  n 

Def.  The  Probability  of  an  event  is  the  mtio  of 
the  chances  which  favor  it  to  the  whole  number  of 
chances  for  and  against  it. 

Illiistraiions.     If  an  event  is  certain,  its  probability  is  1. 
If  the  chances  for  and  against  an  event  are  even,  its  prob- 
ability is  -,' 

If  an  event  is  impossible,  its  probability  is  0. 

Cor.  1.  If  the  probability  that  an  event  will  occur 
is^?,  the  probability  that  it  will  fail  is  1  —  2^- 

Cor.  2.  A  probability  is  always  a  positive  fraction, 
greater  than  0  and  less  than  1. 

2(U>.  Mrlhodof  Prohahililics.  To  find  the  probability  of 
an  event,  we  must  be  able  to  do  two  things : 


^:l' 


• 


!■      ' 


.*" 


^i* 


;  ti , , 


300 


PUOBMilLlTIEH. 


\ 


l^' 


*  f 


...  i ,  si 


'm 


1.  Eninn crate  all  possihle  ways  in  irhicJi  the  event 
Tuaij  occur'  or  fail,  it  hciiii>  supposed  tJiat  these  waija 
are  all  equally  /rrobabJc. 

2.  iJetcnrlfie  hnir  many  of  these  ivays  u'ill  lead  to 
the  event. 

If  n  be  the  total  number  of  ways,  and  m  the  number  which 

1)1 
lead  to  the  event,  the  probability  required  is  —  • 

EXERCISES. 

1.  A  die  has  2  white  and  4  black  sides.  What  is  the  prob- 
ability of  throwing  a  white  side  ? 

2.  A  bag  contaiup  n  balls  numbered  from  1  to  w,  the  even 
numbers  being  while  and  the  odd  ones  black.  What  is  the 
probability  of  drawing  a  l)lack  l)all  when  n  is  an  odd  number? 
What,  when  n  is  an  even  number  ? 

3.  A  bag  contains  3;i  +  2  balls,  of  which  numbers  1,  4,  7, 
etc.,  are  white  ;  2,  5,  8,  etc.,  are  red;  3,  G,  9,  etc.,  are  black. 
What  are  the  respectiye  i)robabilities  of  drawing  a  white,  red, 
and  black  ball  ? 

Rem.     In  the  last  example  the  probabilities  are  all  less  than  ^  ;  thcro- 

fore,  shoiiid  one  attempt  to  guess  the  color  of  the  ball  to  be  drawn,  Ik; 
would  be  more  likely  to  be  wrong  than  right,  no  matter  what  color  lie 
guessed.  This  exemi/iifies  a  lesson  in  practical  judgment  to  be  drawn 
from  the  theory  of  probabilities.  If  there  are  three  or  more  possible  re- 
sults of  any  cause,  it  may  happen  that  the  best  judgment  would  be  more 
likely  to  be  wrong  than  right  in  attempting  to  predict  the  result.  Thus, 
if  there  are  three  presidential  candidates  with  nearly  equal  chances,  the 
chances  would  be  agrinst  the  election  of  any  one  that  might  be  named. 

Gamblers  of  the  turf  are  nearly  alwavs  found  betting  odds  against 
every  horse  that  may  be  entered  for  a  race,  though  it  is  certain  that  one 
of  them  will  win. 

Hence,  if  a  natural  event  may  arise  from  a  number  of  causes  with 
nearly  equal  facility,  it  is  unphilosophical  to  have  any  theory  whatever 
of  the  cause,  because 'the  chances  may  be  against  the  most  prohaI)le 
cause  being  the  true  one. 

Probabilities  clependiiif?  upon  Coiiibiiiatioiis. 

267.  Prohlem  i.  Two  coins  are  thrown.  What  arc  the 
respective  probabilities  that  the  result  will  be  :  Both  heads? 
head  and  tail?  both  tails? 


PROBABILITIES. 


301 


he  event 
,ese  u'dijii 

[I  lead  to 
.ibcr  wlilcli 


s  the  prob- 

n,  the  even 
Vhat  is  the 
(1  number? 

)crs  1,  4,  7, 
,  are  bltick. 
L  white,  red, 


han  2 ;  there- 


of causes  with 

"ovy  wliatt'vn- 

liiost  probal)le 


At  first  siglit  it  miglit  uppeiir  that  tiie  cliances  in  favor  of 
these  tliree  r  :ults  were  equal,  and  tliat  therefore  the  probabil- 
ity of  each  was     '    But  this  would  be  a  mistake.     To  find  the 
o 

probabilities,  we  must  combine  the  possible  throws  of  the  iirst 
(•(•in  (whicli  call  A)  with  the  possible  throws  of  the  second 
(which  call  B),  thus  : 

A,  head  ;  B,  head. 

A,  head  ;  B,  tail. 

A,  tail  ;  B,  head. 

A,  tail ;  B,  tail. 

These  combinations  are  all  cqnally  probable,  and  while 
tliere  are  only  one  each  for  both  heads  and  both  tails,  there  are 

two  for  head  and  tail.     Hence  the  probabilities  are    . ,  -,  -v* 

The  sum  of  these  three  probabilities  is  1,  as  it  ouglit  always 
to  be  when  all  possible  results  are  considered. 

Proh.  2.  Five  coins  are  thrown.  What  are  the  respective 
probabilities:         o  heads,  5  tails? 

1  head,  4  tails? 

2  heads,  3  tails? 
etc.  etc. 

Let  the  several  coins  be  marked  a,  h,  c,  (h  p.  Coin  a  may 
be  either  head  or  tail,  making  two  cases.  Each  of  these  two 
cases  of  coin  a  may  be  combined  with  either  case  of  b  (as  in  the 
last  example),  making  4  cases. 

Each  of  these  4  cases  may  be  combined  with  either  case  of 
coin  c,  making  8  cases. 

Continuing  the  process,  the  total  number  of  cases  for  five 
coins  is  2^  =  32. 

Of  these  32  cases,  only  one  gives  no  head  and  5  tails. 

There  are  5  cases  of  1  head,  namely :  a  alone  head,  ^  alone 
head,  etc.,  to  c. 

2  heads  may  be  thrown  by  coins  a,  b;  a,  c,  etc. ;  b,  c  ;  b,  d, 
etc.;  c,  d,  etc. ;  that  is,  by  any  combination  of  two  letters  out 
of  the  five,  a,  b,  c,  d,  e.    Hence  the  number  of  cases  is 

ri  =  10. 


I  > 


I   I 


,   I 


•  9 


11 


lii 


\l 


TT 


802 


rnor,AiiiLirii^:8. 


%:     . 


n\ 


>   I 


Pi 


In  the  same  way  the  minibt'i-  of  cases  corresponding  to  3, 
4,  and  5  heads  are,  respectively, 


Cl  =  10,         6't  =  5, 


Cl  =  1. 


Dividing:  })y  the  whole  number  of  cases,  we  Hnd  the  respec- 
tive probabilities  to  be 

J^^  5       K)^         10       5        1 

3^'        3:e'    JW'        iW'     32'     3^' 

The  following  general  proposition  is  noAv  to  be  proved  hv 
the  student : 

Theorem.  //  thei^e  are  n  coins,  the  probability  of 
throwing  s  heads  and  n  —  s  tails  is 

2«" 

From  this  result  we  may  prove  the  theorem  in  combimi- 
tions  of  §  2G2.  If  we  suppose,  m  succession,  5  =  0,  5  =  1, 
s  =  2,  etc.,  to  s  =  n,  the  respective  probabilities  of  0  head, 
1  head,  2  heads,  etc. ,  will  be 


cl 


2 


n 


2n' 


O  2 

2 


71 


etc.,     to 


2n' 


Because  the  sum  of  all  these  probabilities  must  be  unitv, 
we  find 

Proh.  3.  Two  dice  are  thrown  at  backgammon.  What  are 
the  respective  probabilities  of  throwing  5  and  G  and  two  6's? 

If  we  call  the  dice  a  and  h,  any  number  from  1  to  6  on  n 
may  be  combined  with  any  number  from  1  to  6  on  h.  There- 
fore, there  are  in  all  3G  possible  combinations. 

In  order  to  throw  two  6's,  a  must  come  6  and  h  al^D. 
Therefore  there  is  only  one  case  for  this  result,  so   that  its 

probability  is  ^-  • 

To  bring  5  and  G,  a  may  be  ,5  and  h  G,  or  h  5  and  a  6.  So 
there  are  two  cases  leading  to  this  result,  and  its  probability  is 

?'   -  L 
30  ~  18* 


PROBABILITIES. 


303 


Note.  That  5  and  G  are  twice  as  i)robiible  ns  a  douM*'  0  may  be 
ck'arly  neon  by  supposing  that  the  two  dice  are  tlirown  in  suoceMsion.  If 
the  first  tlirow  is  eitlier  5  or  (>,  tlicro  is  a  cliancc  for  tlic  combination  5,  0, 
but  there  is  no  chauce  for  a  double  (>  unh:ss  thi;  lirst  llirow  is  0. 

Prob.  4.  If  three  dice  ure  tlirown,  what  are  the  respective 
probiibihties  that  the  nunibcrs  will  be: 

The  sohition  of  this  case  is  left  as  an  exercise  for  the 
.student. 

Prob.  5.  From  a  bag  containing  3  wliitc  and  2  lilack  l)alls, 
•^  ijalls  are  drawn.     Wiiat  are  tlic  respective  probabihties  of 

Both  balls  white? 
1  white  and  1  black  ? 
Both  black  ? 

Since  any  3  balls  out  of  5  may  be  drawn,  the  total  numlior 
of  eases  is  Cg. 

Only  one  of  these  combinations  consists  of  tAvo  white  lialls. 

C-i  of  the  cases  bring  both  l)alls  black. 

A  white  and  Idack  are  formed  by  combining  anyone  of  the 
three  white  with  any  one  of  the  two  black. 

The  respective  probabilities  can  now  be  deduced  by  the 
student. 

EXERCISES. 

1.  It  takes  two  keys  to  unlock  a  safe.  They  are  on  a 
bunch  with  two  others.  The  clerk  takes  three  keys  at  random 
from  the  bunch.  What  is  the  probability  that  he  has  both  the 
safe  keys? 

2.  A  x)arty  of  three  persons,  of  whom  two  are  brothers,  seat 
llieinselves  at  random  on  a  bench.  What  are  the  ])robabilities 
('')  that  the  brothers  will  sit  together,  (b)  that  they  will  have 
the  third  man  between  them  ? 

3.  If  two  dice  are  thrown  at  backgammon,  what  are  the 

jn'obabilities 

(a)  Of  two  aces  ? 

(b)  Of  one  ace  and  no  more  ? 

4.  In  order  that  a  player  at  backgammon  may  strike  a  cer- 


>i 


I',   'r 


\  ; 


r; 


■l.' 

1 

( 

\ 

t     1 

(      1 

1  'I    ' 


t 

I  .       I 


304 


PnoiiABlJATlKS. 


I** 


} 

i ;  , 


'•i: 


'.f 

t 


H.    ' 


!-; 


*J: 


^ 


tuiii  point,  the  sum  of  the  mim})ers  thrown  must  be  8.     What 
are  his  chances  of  succeeding  in  one  throw  of  his  two  dice  ? 

5.  A  ptirty  of  lU  persons  sit  ut  a  round  table.  Wiuit  is  I'lc 
probubiUly  that  ]\Ir.  Taylor  and  Mr.  Williams  will  bu  next  k. 
each  other?     (See  §  ^5;J.) 

6.  An  illiterate  servant  puts  I  wo  works  of  2  volumes  endi 
ui)on  a  siielf  at  random.  What  is  the  probability  that  both 
pair  of  companion  volumes  are  together? 

7.  A  gentleman  having  three  i)air  of  boots  in  a  closet,  scut 
a  blind  valet  to  bring  him  a  pair.  The  valet  took  two  boots  at 
random.  What  are  the  chances  that  one  was  right  and  tln' 
other  left  ?    What  is  the  probiibility  that  they  were  one  pair? 

8.  If  the  volumes  of  a  J3- volume  book  are  placed  at  rjinddiii 
on  a  shelf,  what  is  the  i)robability  that  they  will  be  in  regular 
order  in  either  direction  ? 

9.  A  man  wants  a  particulpr  span  of  horses  from  a  stud 
of  8.  llis  groom  ])rings  him  5  horses  taken  at  random.  What 
is  the  probability  that  both  horses  of  the  span  are  amongst 
them  ? 

10.  From  a  box  containing  5  tickets,  numbered  1  to  5, 
3  are  drawn  at  random.  What  is  the  probability  that  numbers 
3  and  5  are  both  amongst  them  ? 

11.  The  same  thing  l)cing  supposed,  what  is  the  probability 
that  the  sum  of  the  two  numliers  remaining  in  the  box  is  C  ? 

12.  Of  two  purses,  one  contains  5  eagles  and  another  in 
dollar-pieces.  If  one  of  the  purses  is  selected  at  random,  .iiul 
a  coin  taken  from  it,  what  is  the  probability  that  it  is  an 
eaefle  ? 


13.  From  a  bag  containing  3  white   and   4  black 


Dal 


5   i      9 


3  balls  are  drawn.     What  is  the  probability  that  they  arc  of 
the  same  color  ? 

14.  The  better  of  two  chess  pla3Trs  is  twice  as  likely  to  win 
as  to  be  beaten  in  any  one  game.  Wh;it  chance  has  his  weaker 
opponent  of  winning  3  games  in  a  match  of  3  ? 

15.  From  a  bag  containing  m  white  and  n  black  balls,  two 
balls  are  drawn  at  random.  What  is  the  probability  that  one 
is  white  and  the  other  black  ? 


rJiOJJAJJI/JTIhU 


305 


arc  amon;rst 


1 6.  From  a  bag  coiilaiiiing  1  white,  2  red,  and  3  l)la('k 
bulls,  3  balls  aro  drawn.  WliaL  is  the  probability  that  tlu'v  am 
nil  of  ditrerent  colors  ? 

17.  If  n  coins  are  thrown,  what  is  the  chance  that  there 
will  be  one  head  and  no  more  ? 

18.  From  a  Congressional  committee  of  0  Iicpublicans  and 
0  Democrats,  a  snb-eommittee  of  3  is  chosen  by  lot.  What,  is 
the  probability  that  it  will  be  composed  of  two  Kepublicans 
and  one  Democrat  ? 

Coiiipouiid  Events. 

208,  Theorem  I.  The  prohfthilifi/  thrtt  hro  indrpcjid- 
riit  events  will  both  lutppeii  is  e(pi<tl  lo  llie  ])V(Hhiet  of 
tlirir  separate  prohahilities. 

Proof.  For  the  lirst  event  let  there  bo  m  cases,  of  which 
y/  arc  favorable;  and  for  the  second  w  cases,  of  which  q  are 
liivoraljic.     Then,  by  definition,  the   respective   probabilities 

will  be     -  and  -• 
m  n 

When  both  events  are  tried,  any  one  of  the  m  cases  may  bo 
combined  with  any  one  of  the  qi  cases,  making  in  all  in  x  n 
cnnibinatious  of  eqnal  probability. 

The  combinations  favorable  to  both  events  will  be  those 
only  in  which  one  of  the  p  cases  favorable  to  the  first  is  com- 
liiiiod  with  one  of  the  (j  cases  favorable  to  the  second.  The 
niinil)er  of  these  combinations  is  p  x  (p 

Therefore  the  probability  (hat  both  events  will  happen  is 

Q 


m  X  II       III 


?i' 


which  is  the  prodnct  of  the  individual  probabilities. 

If  there  arc  three  events  of  which  the  probabilities  are  p,  q, 
and  r,  and  we  Avish  to  find  the  probability  that  all  three  will 
happen,  we  may  by  what  precedes  regard  the  concurring  of  the 
rivst  two  events  as  a  single  event,  of  which  the  probability  is 
pq.  Then  the  probability  that  the  third  event  m\\  also  con- 
cur is  the  product  of  this  probability  into  r,  or 


pqr. 


i 


't 


f ' 


'1 


1M 


20 


it 


'  IT 


% 


yoC) 


PiiOliAlillJTlF.S. 


rrocTcdiii;^  in  tliu  tjauic  way  willi   1,  .">,  (I,  ....  events,  wo 

rciuli  the  ^a'licnil 

'I'liKoiuiM  II.  Tlic,  i>r()hahnitii  flutt  ni/f/  number  of  in- 
(Ir/trmirnl  crrntu  will  (ill  orr/ir  in  c(/iutl  to  the  etHitinKetl 
in'iidiiet  of  their  indiridital  fir<t/)((f)i/itie8. 

liii.M.  'riiid  thooivm  is  of  grcut  })nicti('iil  U8o  us  a  giiick-  lo 
our  oxpoc'tiitions.  It  Uaclius  that  if  Hucccsa  in  an  ontcrpriso 
rt'(|niros  the  concurrence  of  a  givut  nunihcr  of  favorable  cir- 
cumstances, the  chances  nniy  he  greatly  against  it,  altiioiigii 
eacii  circumstance  is  more  likely  than  not  to  occur. 

This  is  illustrated  by  the  following 

ExAMi'LH  I.  A  traveller  on  a  journey  by  rail  has  8  connec- 
tions to  make,  in  order  that  lie  may  go  through  on  time. 
There  arc  two  chances  to  one  in  favor  of  each  connection. 
What  is  the  probability  of  his  keeping  on  time  ? 


The  probability  of  each  connection  l)cing  „  ,  the  probabil- 
ity  of  successfully  making  the  first  two  connections  will,  by  the 

.J  ,  the  first  three  (^) ,  and  all  eight 

Therefore  there  are  ^5  chances  to  1  iigainst  his  going 
through  on  time. 

On  the  other  hand,  if,  instead  of  any  one  accident  being 
fatal  to  success,  success  can  be  prevented  only  by  the  concur- 
lence  of  a  series  of  accidents,  the  probability  of  failure  ui:iy 
become  verv  small. 

Ex.  2.     A  shiji  starts  on  a  voyage.     It  is  an  even  chance 

that  she  will  encounter  a  heavy  gale.     The  probability  that 

9 
she  will  not  spring  a  leak  in  the  gale  is  ^ri'    If  ^  ^^^^  occurs, 

9 

there  is  a  probability  of  jr-  that  the  engine  will  be  able  to 


pump  her  out.     If  they  fail,  the  probability  is  -r  that  the  coni- 


r  1. . , 


PliOBAUlLlTlKS.  307 

parlincnls  will  keep  (lie  slilp  alloal.  If  mIk'  .siiikn,  it  is  lui  I'vun 
clKiiiL't'  timt  any  one  piissi'ii^^^cr  will  l»o  saved  by  the  hoatn 
Wliiit  is  the  prohability  lliat  uny  iiiUi vicinal  passenger  will  bo 
lufii  at  aeu? 

The  probability  tluit 

tln'  sliip  will  inei't  a  heavy  gale  is 

m 

tlic  ship  will  Spring  a  leak  in  (be  gale  is ~ 

thc  engines  cannot  pnnip  her  ont  is — 

the  compartments  cannot  keep  her  ulloat  is 

4 

tiu'  bouts  cannot  save  the  passenger  is - 

Tlie  CMitinncd  product  of  these  probabilities  is  y.v* 
which  is  the  probability  that  ti'e  passenger  will  be  lost. 

'^01).  The  preceding  tlieorem  as  enunciated  supposes  that 
the  several  events  are  indcjKinilody  that  is,  that  the  prol>al)ility 
(il'thj  occurrence  of  any  one  is  not  affected  by  the  occurronco 
or  non-occurrence  of  the  others.  To  investigate  what  moditi- 
cation  is  required  when  the  occurrence  of  one  of  the  events 
alters  the  probability  of  another  of  the  events,  let  us  distinguish 
the  two  events  as  the  first  and  second.     \Vc  then  reason  thus : 

Let  the  total  nnmber  of  equally  possible  cases  bo  ?/?,,  and  let 
/)  (tf  these  cases  favor  the  first  event.     Its  probabiHty  will 

then  be   — • 
m 

It  is  certain  that  the  events  cannot  both  happen  unless  the 

first  one  happens.     Hence  the  cases  which  favor  both  events 

can  be  found  only  among  the  p  cases  which  favor  the  first. 

Let  q  of  these  p  cases  favor  the  second  event.     Then  the  prob- 

ability  of  both  events  will  be  —  ■ 

m 

In  case  the  first  event  happens,  one  of  the  p  cases  which 


I  ► 


308 


PROBABILITIES, 


V. 


I!  . 
t 


^ 


4 


''^h 


fiivor  it  must  occur,  and  the  i)robiibility  of  the  second  event 
will  then  be     •     Then 


Pro])iibility  of  both  events  =  —  =  —  x     • 

•^  III  711         ]) 


Hence, 


Theorem.  The  prohahil Ity  fJiat  tiuo  events  will  both. 
occur  is  equal  to  the  jn'ohdhilitij  of  the  first  event  iiiu/li- 
plied  by  the  probability  of  the  second,  in  case  the  Jir4 
occurs. 

By  continuing  the  reasoning  to  more  events,  we  reach  the 
general 

Theorem.  Tlie  probabiliti/  that  a  number  of  events 
will  all  occur  is  equal  to  the  product 

i  X  Prob.  of  second  in  case  first  occurs. 
Prob.  of  first  \  x  Prob.  of  third  in  case  first  two  occur. 

(   X  Prol).  of  fourth  in  case  first  three  occur, 
etc.  etc.  etc. 

Example.  From  a  bag  containing  2  white  and  3  black 
balls,  2  balls  are  drawn.  What  are  the  probabilities  (1)  that 
both  balls  are  white,  (2)  that  both  are  black? 

This  problem  has  already  been  solved,  but  we  are  now  to 
see  how  the  answers  may  be  reached  by  the  last  theorem.  It 
is  evident  that  we  may  suppose  the  two  balls  drawn  out  one 
after  the  other,  and  the  probabilities  of  their  being  Avhitc  or 
black  will  be  the  same  as  if  both  were  drawn  together. 

I.  Both  balls  white.     The  probability  that  the  first  liall 

2 

drawn  is  white  is  -•     If  it  really  proves  to  be  white,  there  will 

o 

be  left  1  white  and  3  black  balls.     In  this  event,  the  probability 
that  the  second  also  will  be  white  is  - 


11 


Hence  the  probability  that  both  are  white  is 

2       1  _  ^ 
5  ^  4  ""  10* 


PROBABILITIES. 


:j()0 


'i 


second  event 


II.  Both  halls  Mack.    Ai)i)lyiiig  the  same  rcasoniug,  we 
fiud  fur  the  probability  of  ihis  case, 


3       1   _   '3 
5  ^  2  ~  l6' 


EXERCISES. 

1.  Two  men  embark  in  separate  commercial  enterprises. 
Tlie  odds  in  favor  of  one  are  3  to  2;  in  favor  of  the  otlier,  2 
to  1.  What  arc  the  probabilities  (1)  that  both  will  sncceed? 
{2)  that  both  will  fail? 

2.  The  probability  that  a  man  will  die  within  ten  years  is 

,,  and  that  his  wife  will  die  is  --•     What  are  the  respective 
probabilities  that  at  the  end  often  years, 

(«)  Both  are  living? 

((3)  Both  are  dead  ^ 

(y)  Husband  living,  l)ut  wife  dead? 

(6)  Husband  dead,  but  wife  living? 


3.  The  probability  that  a  certain  door  is  locked  is  ;. 


2 


3 


The 
and 


key  is  on  a  bunch  of  -1.     A  man  takes  2  of  the  four  key.- 
goes  to  the  door.     What  are  the  chances  that  he  will  be  able  or 
unable  to  go  through  it? 

4.  Two  bags  continn  each  4  black  and  3  white  l)alls.  A 
]icrson  draws  a  ball  at  random  from  the  first  bag,  and  if  it  ]ie 
^vhite  he  puts  it  into  the  second  bag,  mixes  the  balls,  and  then 
draws  a  ball  at  random.  What  is  the  prol)ability  of  drawing 
a  white  ball  from  each  of  the  l)ags  ? 

5.  If  a  Senate  consists  of  ni  Democrats  and  n  Republicans, 
Avhat  is  the  probability  that  a  committee  of  three  will  include 
'I  Democrats  and  1  Republican  ? 

6.  A  bag  contains  2  white  balls  and  5  black  ones.  Six 
]H'ople,  A,  B,  C,  D,  E,  F,  are  allowed  to  go  to  the  bag  in  alpha- 
betical order  and  each  take  one  ball  out  and  keep  it.  The 
first  one  who  draws  a  white  ball  is  to  receive  a  prize.  What 
are  their  respective  chances  of  winning? 

Note.  A's  chance  is  easily  calculated,  because  lie  haa  the  draw  from 
all  7  balls. 


It   » 


I         ! 


1  '     ' 

'  I     i 

.i   i 


"mm 


310 


PROBABILITIES 


^« 


.^f'' 


V     "     If! 


\<  if 

I 


lih.Ii:; 


li.^^^ 


Ilk 


i«^^ 


In  ordor  that  B  may  win,  A  must  first  fail.  Therefore,  to  find  B'h 
probability  we  find  (1)  the  probability  that  A  fails,  (2j  the  probability  that 
if  A  I'uila  then  B  will  win.  Wo  then  take  the  product  of  these  probabili- 
ties. 

In  order  that  C  may  gain  the  prize,  (1)  A  must  fail,  (2)  B  must  fail, 
(;))  ( '  himself  iuust  gain.  So  we  find  the  successive  probabilities  of  the:^e 
occurrences. 

Continuing  to  F,  we  find  that  he  cannot  win  unless  the  5  men  before 
liiiu  all  miss.  lie  is  then  certain  to  gain,  because  only  the  two  white 
b;'.l!;s  would  be  left. 

7.  Two  men  liave  one  throw  oach  of  a  coin.  X  offers  a 
prize  if  A  throws  head,  and  if  he  fiiils,  but  not  otherwise,  B 
nijiy  try  for  tlie  prize.  If  both  fail,  X  keeps  the  prize  himself. 
Whut  are  the  respective  chances  of  the  three  men  having  the 
prize  ? 

8.  A  and  B  are  alternatelv  to  throw  a  coin  until  one  of 
them  throws  a  head  and  becomes  the  winner.  If  A  has  the 
first  throw,  what  are  tlieir  respective  chances  of  winning? 

9.  A  crowd  of  n  men  are  allowed  to  throw  in  the  same  way 
for  a  prize,  in  alphabetical  order,  the  game  ceasing  as  soon  as  a 
head  is  thrown.  What  are  the  respective  chances  of  the  con- 
testants ? 

10.  Three  men  take  turns  in  throwing  a  die,  and  he  wlio 
first  throws  a  G  wins.     What  are  their  respective  chances? 

11.  If  -I  cards  are  drawn  from  a  pack  of  52,  show  that  the 
probability  that  there  will  be  one  of  each  of  the  four  suits  is 

39  26  13 
5l'50'49' 

12.  One  purse  contains  5  dimes  and  1  dollar,  and  another 
contains  G  dimes.  5  pieces  are  taken  from  the  first  purse  and 
put  into  the  second,  and  after  being  mixed  5  are  taken  from 
the  second  and  put  into  the  first.  Which  purse  is  now  most 
likely  to  contain  the  dollar  ? 

13.  Of  two  purses,  one  contains  4  eagles  and  2  dollars,  the 
other  4  eagles  and  6  dollars.  One  being  taken  at  random,  and 
a  coin  drawn  from  it,  what  are  the  respective  probabilities 
that  it  is  an  eagle  or  a  dollar? 


PROBABILITIES. 


311 


Cases  of  Unequal  Probability. 

270.  Dff.  If  two  or  more  possible  events  fire  so 
reltit(*d  that  only  one  of  tlieni  can  happen,  tliey  are 
called  Mutually  Exclusive  Events. 

TfiEOREM.  The  prohahility  tJutt  some  one  of  several 
ixehtsive  events,  we  care  not  ivhieh,  will  occur,  is  equal 
Id  the  sum  of  their  separate  prohahilities. 

Proof.  Let  tliGro  be  m  possible  and  equally  probable  cases 
in  all ;  let  p  of  these  cases  be  favorable  to  one  event,  q  to  the 

P        (I        T 

second,  r  to  the  third,  etc.,  so  that  — ,  — .  — ,  are  the  re- 
>pective  probabilities. 

Since  only  one  of  the  events  is  possible,  the  p  cases  which 
favor  one  must  be  entirely  different  from  the  q  cases  wliicli 
fiivor  the  second,  and  these  cases  'p-\-q  must  be  entirely  differ- 
ent from  the  r  Avhich  favor  the  third,  QiQ. 

Hence  there  will  be  7; -}- ^  +  r  +  etc. ,  cases  which  favor  some 
oii(>  or  another  of  the  events.  Hence  the  probability  that  some 
one  of  these  events  will  occur  is 

;>  +  7  +  ^'  +  otc. 

which  is  equal  to  the  sum  of  the  probabilities, 

P       q        r 

-^  +  —  H +  etc. 

w       m      m 

RE^^.  If  the  concurrence  of  some  two  events,  say  the  first 
niitl  second,  had  been  possible,  some  one  or  more  of  the  p  cases 
which  favor  the  first  would  have  been  found  among  the  q  cases 
which  favor  the  second.  Then  the  whole  number  of  cases 
which  favored  either  event  would  have  been  less  than  p-\-q, 
and  the  probability  that  one  of  tiie  two  events  would  happen 
li'ss  than  the  sum  of  their  respective  probabilities. 

271.  General  Problem.  To  find  the  probability  that 
an  event  of  luhicli  the  probability  on  any  one  trial  is  p, 
will  happen  erectly  s  times  in  n  trials. 


k»' 


S     t 


t   ■  \ 


A 


'»! 


,■-,-.-, 


?12 


rROUAniLITTES. 


k..P 


r       : 


m' 


( I 


M 


Tliis  problem  is  at  the  V,[xd\i  of  some  of  the  widest  apph'ca- 
tioiis  of  the  theory  of  iirol)ability  to  i)ractical  questions,  espe- 
cially those  associated  with  life  and  lire  insurance.  The  con- 
ditions which  it  implies  are  therefore  to  be  fully  comprehended. 

Wc  may  conceive  a  trial  to  mean  giving  the  event  an  opj^or- 
tunitg  to  happen.  The  simplest  kind  of  trial  is  that  of  throw- 
ing a  coin  or  die.  At  each  tiirow,  any  side  has  an  opportunity 
to  come  up.  Then,  if  we  throw  50  ])ieces,  or  Avhicli  amounts 
to  the  same  thing,  throw  the  same  piece  50  times,  there  will 
be  50  trials;  and  we  may  iiupiire  into  the  probability  that  a 
given  side  will  be  thrown  exactly  9  times  in  these  trials. 

The  same  conception  occurs  in  another  form  if  we  have  50 
men,  each  of  whom  has  an  equal  chance  of  dying  within 
6  years.  Waiting  to  sec  if  any  one  man  will  die  in  the  course 
of  ^\^  5  years  is  a  trial,  so  that  there  are  50  trials  in  all,  and 
wc  may  inciuire  into  the  probability  that  9  of  the  men  will  die 
during  the  trials,  just  as  in  the  case  of  50  throws  of  a  die. 

Let  us  distinguish  the  several  trials  by  the  letters 

«,       i,        C,       d,       Cy        ....      Uy 

which  must  be  n  in  number. 

1.  In  order  that  the  event  may  not  happen  at  all,  it  must 
fail  on  every  one  of  the  n  trials.  The  probability  of  this 
(§  208,  Th.  II)  is  (1  —p)''.  This  is  therefore  the  probability 
that  it  will  not  happen  at  all. 

Because  the  probability  of  the  event  happening  on  any  one 
trial  is  p,  the  probability  of  its  failing  is  1  —  p.  We  now 
compare  the  possible  results. 

2.  The  event  may  happen  once  on  any  one  of  the  n  trials, 
a,  hy  c,  etc.  In  order  that  it  may  happen  only  once,  it  must 
fail  on  the  other  n  —  1  trials.  The  probability  that  it  will 
happen  on  any  one  trial,  say  e,  and  also  fail  on  the  remaining 
n  —  1  trials  is,  by  the  same  theorem. 

Because  there  are  n  trials  on  which  it  may  equally  happen, 
the  probability  that  it  will  happen  once  and  only  once  is 

np  (1  —  p)^-\ 


PROBABrUTlES. 


313 


3.  The  event  may  luippen  twice  on  any  two  trials  out  of  the 
n  trials.  In  order  that  it  may  happen  twice  only,  it  must  fail 
on  the  otlier  w  —  2  trials.     Taking  any  one  combination,  say 

Happen  on     h,  d; 

Fail  on  a,  c,  e, n, 

the  probability  is  p^{l  —  py^~^. 

But  it  may  happen  twice  on  any  combination  of  two  trials 
out  of  the  71  trials,  a,  b,  c,  .  .  .  .  u.  Because  these  com])in;i- 
tions  are  mutually  exclusive  (§  270),  the  total  probability  of 
liappening  twice  is 

0^2  p'  (1  -  P^''- 

4.  In  general,  in  order  that  the  event  may  happen  Just  s 
limes,  it  must  happen  on  some  combination  of  s  trials,  and  fail 
on  the  complementary  combination  of  n  —  .s-  trials.  The 
probability  on  any  one  combination  is  ;/(!  — p)'^~^  ^i^id  there 
are  6^^*^  such  combinations.  Hence  the  general  probability  of 
happening  s  times  is 

C^  p'^  {1  -  p^-s.  (a) 

If  there  is  on  each  trial  an  equal  chance  for  and  against 
the  event,  then  p  = -.  and  l—p  =  ^'  The  probability  of 
the  event  happening  s  times  then  becomes 

2'^" 

This  case  corresponds  to  that  already  treated  in  §  267, 
Problem  2,  and  the  result  is  the  same  there  found. 

EXERCISES. 

I.  A  die  having  two  sides  white  and  four  sides  black  is 
thrown  5  times.  What  are  the  respective  probabilities  of  a 
white  side  being  thrown  1,  2,  3,  4,  and  5  times? 

Note.    Here  p,  the  probability  of  a  white  side  on  one  throw,  Is  ^ ,  and 

1  —  jt)  =  T  •   The  number  n  of  trials  is  5. 
o 


\      I 


,! 


i   1 


'  it 


314 


PROBADILITIES. 


IV. 


2.  Of  0  healtliy  men  aged  50,  the  probability  that  any  one 
will  live   Lo  80  is   ,  •     What  is  the  i)robtibility  that  three  or 

more  of  them  will  live  to  that  age  ? 

3.  A  chess-player  whose  chances  of  winning  any  one  game 
from  his  opponent  are  as  3  to  1,  undertakes  to  win  3  games 
out  of  4.     What  is  the  probability  that  he  will  be  able  to  do  it? 

Note.  It  would  be  a  fallacy  to  pn<)pose  that  the  probability  required 
is  that  of  winning  exactly  3  games,  because  lie  will  equally  win  if  he 
wins  all  four  games. 

^ZTZ.  Evcnis  of  Afaximiim  Prohahility.  Returning  to  the 
general  expression  {a),  let  us  inquire  what  number  of  time^i 
the  event  is  most  likely  to  occur  on  11  trials.  The  required 
number  is  that  value  of  x  for  which  the  i)robability 

is  the  greatest. 

If  we  call  Pg  tho  pro!;:t,bility  that  the  event  will  happen 
exactly  .v  times,  and  if  s  is  to  be  the  number  for  which  the 
probability  is  gTv-^atest,  we  must  have 

Ps  >   Ps-U 
Ps  >   P.fl. 

Substituting  for  these  quantities  the  corresponding  forms 
of  the  expression  {a),  which  is  equal  to  Ps,  we  have 

C'sp^  (1  -  pY-'  >  cUp^'  (1  -pY-'^\  ^  ,j. 

r;'^;/(i  -pY~'  >  C'snp'^'  (1  -  pY-'-\  ) 

The  general  formula  for  6'«  in  §  257  gives 


C"  = 


^  "I"  1  ri^ 


^s+1  — 


n 


S  ^n 


(c) 


Cl\ 


s  +  1 

Hence  we  have,  by  dividing  both  terms  of  the  first  in- 
equality (b)  by  C's-ip"-^  (1  —pY~^ 

n—s+1     .     . 
p  >  I— p. 


PROBAiniATIES. 


3J5 


Multiplying  by  s,  this  becomes 

np  —  sp  -f-  ^;  >  ,s  —  .^p. 
Interebunging  the  members  and  reducing,  we  have 

.s-  </>(//  +  I).  {(l) 

Now  divide  the  second  inequality  {li)  by  C'^p^  {\  —  pY~^^, 
und  reducing  by  the  second  e(|uation  {c),  we  have 

Multiplying  by  s  +  1  and  reducing,  we  find 

.s-  >  p{n+  1)-1.  (e) 

Comparing  the  ine(iualities  {d)  and  (<),  we  sec  that  ,s  lies 
lietween  the  two  quantities  j)  {n  -\-  1)  and  p  {n  +  1)  —  1; 
that  is, 

,s  ifi  the  greatest  luhole  iiiniihrr  'ui  p  [n  +  1). 

If  the  number  of  trials  )i  is  a  large  numbei-,  and  j)  is  a  snuill 
fraction,  p  (n  +  1)  and  pii  will  differ  only  by  the  fraction  jt. 
We  shall  then  have,  very  nearly, 

,s-  =  pn. 
That  is  : 

I'liKOREM  I.  I7ie  most  pjvhaMe  nuiuher  of  times  that 
(Hi  event  ivill  Jiappen  on  a  great  niunber  of  trials  is  the 
prodact  of  tJis  niunber  of  trials  hy  the  j)robabititij  on 
('((eh'  trial. 

Example.  If  a  life  insurance  company  lias  GOOO  members, 
and  the  i)robability  that  each  member  will  live  one  year  is  on 

the  average  — ,  then  the  most  probable  number  of  deaths 

(luring  the  year  is  100. 

Rem.  It  mast  not  be  supposed  that  in  this  case  the  num- 
ber of  deaths  is  likely  to  be  exactly  100,  but  only  that  they 
will  fall  somewhere  near  it. 

There  is  a  practical  rule  for  determining  what  deviation 
must  be  guarded  against,  the  demonstration  of  which  requires 
more  advanced  mathematical  methods  than  those  employed  in 
this  chapter.     It  is : 


t    ! 


I     t 


•    1 


\ 


n 


,1  i 
■  ii  . 


310 


rnonMni.iTiKs. 


i.  II 


(i  I.' 

I  '    » 


I'll  icoiiEM  J 1.  JJcriatioiis  from  the  vinst  probable  num- 
ber of  (leoths,  etfiKil  to  the  sf/uare  root  of  iJiat  iiinnher, 
will  he  of  fretjueiLt  occiwrence, 

Uevlations  nineh  greater  than  this  sf/uai'e  root  will 
be  of  infrecfneut  oeearrenee,  and  devlatioiis  viore  iJnai 
tu'iee  as  great  will  be  rare. 

Examples.  In  a  coniptiny  of  wliicli  tlie  probable  annual 
number  of  deaths  is  10,  the  actual  number  will  commonly  fall 
between  10—  a/10  and  10  +  VlO,  or  between  7  and  i;j.  1( 
will  very  rarely  happen  that  the  number  of  deaths  is  as  small 
as  4  or  as  large  as  IG. 

If  the  company  is  so  large  that  the  most  probable  number 
of  deaths  is  100,  the  actual  number  will  commonly  fall  betw(  a 
100  —  VlOO  and  100  +  VlOO,  or  between  1)0  and  110. 

If  the  most  i)robable  number  of  desiths  is  1000,  the  actual 
numl)er  will  commonly  range  between  908  and  1032. 

We  now  see  the  following  result  of  this  theorem: 

TJie  'Jrenter  the  nuinhcr  of  deaths  to  he  expected,  the 
greater  will  he  the  proludtle  deviation,  hat  the  less  irill  he 
the  ratio  of  this  deviation  to  the  wh^ole  number  of  deaths. 

Examples.  The  reductions  of-  the  cases  just  cited  are 
shown  as  follows : 


ectod  number 
of  deaths. 

Probahlo 
deviation. 

Ratio  of  deviation 
to  expected  number. 

10 

3 

0.33 

100 

10 

0.10 

1000 

32 

0.03 

Application  to  Life  Iiisiiraiice. 

373.  At  each  age  of  human  life  there  is  a  certain  proba- 
bility that  a  person  will  live  one  year.  This  probability  di- 
minishes as  the  person  advances  in  age. 

It  is  learned  from  observation,  on  the  ])rinciplo  described  in 
the  preceding  section,  that  events  in  a  vast  number  of  trial.-: 
are  likely  to  happen  a  number  of  times  equal  to  the  product  of 
their  probability  on  each  trial,  multiplied  by  the  number  of 
trials. 


PROliMULlTim. 


317 


Therefore,  by  dividlncr  Hu' whole  niinihcr  of  times  the  event 
lias  luippened  by  tlie  wliole  nunibe'r  of  trials,  the  ([uotient  is 
the  most  probable  value  of  the  probability  on  one  trial. 

ExAMPLK.  If  Me  take  50,000  peoi)le  at  the  age  of  25,  and 
reeord  how  many  of  them  are  alive  at  the  end  of  one  year,  this 
is  making  50,000  trials  whether  a  i)erson  of  that  age  will  live 
ohc  year. 

If  41»,G50  of  them  are  alive  at  the  end  of  the  year,  and  350 
are  dead,  we  would  conelude: 

Probability  of  living  one  year,    ....    0.993 
rrobal)ility  of  dying  within  the  year,  .    .    0.007 

The  proba))ility  for  all  ages  may  he  determined  by  taking  a 
great  number  of  infants,  say  1'  00,  and  counting  how  many 
die  in  each  year  until  all  arc  cicad.  If  n  are  living  at  the  age 
?/,  and  )\!  at  the  age  y  -\-  \,  then  the  prol)ability  of  dying 

within  one  year  after  the  age  y  will  be ,  and  that  of 


living  will  be 


n 
n 


n 


It  is  not,  however,  necessary  to  wait  through  a  lifetime  to 
reach  this  conclusion.  It  is  suflticient  to  find  from  observation 
what  proportion  of  the  people  of  each  age  die  during  any  one 
year.  Suppose,  for  instance,  that  the  census  of  a  city  is  taken, 
and  it  is  found  that  there  are  2500  persons  aged  oO,  and  2000 
aged  50.  At  the  end  of  a  year  another  in(iuiry  is  made  to 
ascertain  how  many  are  dead.  It  is  found  that  20  of  the  30 
year  old  people,  and  30  of  the  50  j-ear  old  people  have  died. 
This  would  show: 

At  age  30,  probability  of  dying  within  1  year  =  O.OOS, 
"       50,  ''       "      "  "  "  =  0.015. 

This  same  probability  being  obtained  for  every  year  of  life, 
the  probability  of  living  1  year  at  all  ages  would  be  known. 
Then  a  table  of  mortality  could  l)e  formed. 

A  table  of  mortality  starts  out  with  any  arbitrary  number 
of  people,  generally  100,000,  at  a  certain  age,  fre(juently  10 
years.  It  then  shows  how  many  of  these  people  will  be  living 
at  the  end  of  each  subsequent  year  until  all  are  dead.  The 
following  is  a  specimen  of  such  a  table. 


i!    •• 


I       ! 


I 
■I  i< 


!■ 


'I 


^fe! 


318 


PRoiiAnnjTf/'jfi. 


Tabic  of  3Ioi(jiIitv. 


Prol).  of 

I'n.h.  of 
dying 
witliiii 

Prob.  of 

Prob  of  1 

Agos, 

Living. 

DylriK. 

Hin  viving 

Agos. 

Living. 

Dying,  (surviving 

•lyliii,'  I 
wiiliiii 

a  your. 

the  year. 

u  yc  ur. 

1 

the  J  ear 

lo 

looooo 

442 

.Q955H 

.00442 

60 

58373 

1 

1677  .9vn 

.02873 

1 1 

0.,n,S 

407 

.99:)9l 

.•,0408 

61 

56696 

1760  .9689,') 

.o3 1 04 

13 

iy)\.1\ 

385 

.9961 1 

.oo3S8 

62 

54936 

l84g   .966)4 

.o3.)')5 

iJ 

()Hi(}b 

376 

.99619 

.oo58o 

63 

53087 

i(,36   .96353 

.o36/,6 

14 

9M3yo 

379 

.99614 

.00385 

64 

5ii5i 

2014   .96062 

.03937 

i5 

9^01 1 

396 

.99595 

.00404 

65 

49137 

2080   .95766 

.04233 

i(> 

97()i5 

426 

.99563 

.00436 

I  66 

47057 

21 38  ,  .95456 

.0454  i 

17 

97i,S9 

4')Q 

.99517 

.00482 

67 

449 '9 

2186  1  .95133 

.04866 

IS 

96720 

52  5 

.99457 

.00542 

68 

42733 

2224  '  .94795 

.o52o4 

"> 

9^195 

58 1 

.99396 

.oo6o3 

69 

4o5o9 

2268 

.94401 

.05598 

21) 

9)614 

621 

.99350 

.00649 

70 

38241 

233i 

.93904 

.06095 

■?.\ 

9  ',(;(>3 

645 

.99321 

.00679 

7' 

35910 

240 1 

.93313 

.06686 

2  2 

9434H 

653 

.99307 

.00692 

72 

33509 

2469   .92631 

.07368 

i\ 

93095 

65 1 

.993o5 

.00694 

73 

3 1040 

253 1   .91846 

.08154 

■i\ 

93044 

647 

.99304 

.00695 

74 

28009 

2567  1  .90995 

.09004 

2J 

92397 

647 

.99299 

.00700 

75 

25942 

2.542  :  .9"2oi 

.09798 

■i(-> 

917)0 

65 1 

.99290 

.00709 

76 

23400 

2476   >94i8 

.io58i 

27 

91099 

668 

.99266 

.00733 

77 

20924 

2  i69   .88678 

.1  l32l 

2S 

90431 

686 

.99241 

.00758 

'  7B 

18555 

2247   .87890 

.12109  1 

■J.; 

H9745 

703 

.99216 

.00783  , 

1  79 

i63o8 

2110  j  .87061 

.12938 

3.) 

H9042 

718 

.99193 

.00806  ' 

!  80 

1419^ 

1969   .861 3 1 

.13868 

h 

HSi'j', 

726 

.99178 

.00821 

;  81 

12229 

1823   .85092 

.14907 

ii 

K7)9S 

733 

.99163 

.00836 

1  82 

10406 

1672   .83932 

.  1 6067 

3! 

K6H65 

743 

.99144 

.00855 

1  83 

8734 

1 522  ,  .82573 

.  1 7426 

34 

H6I22 

754 

.99124 

.00B75 

:  «4 

7212 

i36o   .81142 

1 

.18857 

3:') 

85368 

768 

.99100 

.00899 

!  85 

5852 

1186  !  .79733 

,20266 

30 

8/(600 

7H9 

.99067 

.00932 

86 

4666 

1014 

.78268 

.21731 

37 

83811 

811 

.99032 

.00967 

«7 

3652 

849 

.76752 

.23247 

38 

H3ooo 

83o 

.99000 

.01000 

:  88 

2803 

689 

.73419 

.24580 

39 

82170 

844 

.98972 

.01027 

i  ^^ 

2114 

548 

•74077 

.25922 

/.<) 

81 326 

854 

.98949 

.0 1  o5o 

90 

1 566 

',35 

.72222 

•27777 

41 

80472 

860 

.98931 

.01068 

91 

ii3i 

336 

.70291 

.29708 

A2 

79612 

869 

.98908 

.01091 

92 

795 

247 

.68930 

.3 1 069 

4  3 

78743 

888 

.98872 

.01127 

93 

548 

181 

.66970 

.33029 

4i 

77855 

913 

.98827 

.01 172 

94 

367 

i3i 

.643o5 

.35694 

4  5 

76942 

948 

.98767 

.01232 

95 

236 

86  i  .63559 

.36440 

4^3 

75994 

989 

.98698 

.oi3oi 

96 

i5o 

56  ''    .62666 

.37333 ! 

47 

75oo5 

1029 

.98628 

.01371 

97 

?^ 

44  i  .53191 

.46808  1 

4-3 

73976 

1 067 

.98557 

.01442 

98 

33   .34000 

.66000 

49 

72909 

1 102 

.98488 

.oi5i  I 

99 

<7 

II  i     ^3 

hi 

5o 

71807 

ii33 

.98422 

.01577 

100 

6 

4  '        M 

?i 

5i 

70674 

1 167 

.98348 

.0 1 65 1 

101 

a 

2   

52 

69507 

1204 

.98267 

.01732 

102 

0 

53 

54 

683o3 
67052 

I  25 1 

i3o4 

.98168 
.98055 

.oi83i 
.01944 

Note.  The  above  table  is 

that  of 

55 

6574H 

i358 

.97934 

.02065 

the 

English  Ins^titute  of  Act 

laries, 

56 

57 
5H 

64890 
62976 
6 1 5o5 

1414 
147' 
i53i 

.97804 
.97664 
.97510 

.02195 
.02  335 
.02489 

pre 
the 

pared  between  1862  and  186£ 
contiuued  experience  of  t 

»,  from 
wenty 

59 

59974 

i6oi 

.97330 

.02669 

lea 

ding  life  insurance  compan 

iee. 

piKniAiiiLiriKs. 


WW) 


.of 

Prob  (if 

1  1 

<  1  \'  1 1 1  ij 

■in;,' 

uhliiii 

it*. 

the  jcitr. 

'n 

M'i^ri 

93 

.o3 1 04 

J4 

.o33'):j 

53 

.03646 

6a 

.03937 

66 

.042  J  3 

56 

.0454 1 

33 

.04H66 

95 

.o52()i 

UI 

.0559H 

04 

.0609) 

i3 

.o66S^ 

3i 

.0736S 

46 

.oHiiVl 

95 

.09004 

01 

.0979S 

i8 

.loJHi 

78 

.Il32l 

90 

.121 09 

61 

.1293^^ 

3i 

.13868 

52 

.I49»7 

32 

.16067 

73 

,17426 

\2 

.188J7 

33 

.20266 

)8 

.21731 

J2 

.23247 

'9 

.245S0 

77 

.2:5922 

!2 

•27777 

)I 

.29708 

Jo 

.31069 

70 

.33i)29 

)5 

.35694 

'9 

.36440 

)6 

.37333 

)i 

.46808 

)0 

.66000 

is 

that  of 

\ct 

laricsi, 

186C 

»,  from 

if  t 

wenty 

lan 

iee. 

riU)MF,i:M.  To  jitid  ihf  probability  tlmi  ti  person  of  jigc  (t 
will  live  to  tigu  ij. 

Solution.  We  lake  fVoin  tlic  tahle  the  mmihor  liviii;,'  at 
age  //,  and  divide  it  by  tlie  number  living  at  age  a.  Tbc  (|ii(»- 
tieiit  is  the  ju'obabiiity. 

*^^74.  The  principle  on  wiiiei)  the  value  of  a  contiiigcnt 
l)ayment  is  determined  is  the  following: 

TliKOUEM.  The  I'dliir  of  <(  prohahlr  /xn/niritf  is  rf/nttf 
l(t  tlir  stun  to  he-  paid,  inulti/dicd  hij  the  /yn^f/tttji/i/i/  that 
it  will  he  paid. 

Proof.  Let  there  be  n  men,  for  each  of  whom  there  is  a 
probability />  that  he  will  receive  the  sum  s.  Then  by  §  ^72, 
Th.  J.,pn  of  the  men  will  [jrobably  receive  the  i)ayment,  so  that 
the  total  sum  which  all  will  receive  will  probably  be  p/is.  Xow, 
l)efore  they  know  who  is  to  get  the  money,  the  value  of  each 
one's  share  is  eriual.  Therefore,  to  find  this  value,  M'c  divide 
the  whole  amount  to  be  received,  namely,  pns,  by  the  number 
of  men,  n.  This  gives  ps  as  the  value  of  each  one's  chance, 
which  proves  the  theorem. 

Note.  In  this  proof  it  is  tacitly  supposed  that  the  pus 
dollars  are  as  valuable  divided  among  the  p?i  men  as  divided 
among  all  n  men.  But  this,  though  sui)posed  in  mathematical 
theory,  is  not  morally  true.  ^lorally,  the  money  will  do  more 
good  when  divided  among  all  the  men  than  when  divided 
among  a  portion  selected  by  chance.  All  gaml)ling,  whether 
by  lotteries  or  games  of  chance,  is  in  its  total  effects  njjon  the 
pecuniary  interests  of  all  })arties  a  source  of  positive  disadvan- 
tage. This  disadvantage  is  treated  mathematically  by  more 
advanced  methods  in  special  treatises. 

EXERCISES. 

I.  Find  from  the  table  the  probabilities  that  a  person 


a. 

Aged 

6[) 

will   J 

ive   tf 

)   vu. 

b. 

<i 

30 

a 

(. 

80. 

c. 

(( 

50 

a 

a 

60. 

d. 

(( 

60 

i  i 

ii 

70. 

i  •' 


If  ' 

1      I 


M! 


■I  » 
I 


f  !' 


:    i 


h 


1 11 


I!!   i 

ill ' 

I 


'&20 


riioiiMiii.irirs. 


»• 


% 


e.  Aged  70  will  live   to  80. 

/.         "  80  "  **        90. 

//.         '*  90  **  "        95. 

)/.         *'  9')  ♦*  ♦'      100. 

2.  Wliiit  a^jc  is  lliiit  lit  which  it  i.s  an  evuii  (.'ImncT  whi'thcr 
a  |)('rsi»M  i\<xv{\  40  will  be,  living  or  deuil  ? 

3.  Show  Ihat  the  piohahility  that  a  pcrHoti  a<,'('(l  '{()  will  live 
to  TO  is  C(jiial  to  the  j)ro(lu('tor  the  i)rohal)ility  that  he  will  live 
to  00  multiplied  by  the  proluibility  that  11  man  agetl  GO  will 
live  to  70.     (Ai>i)ly  the  theorem  of  §  :*(;9.) 

4.  What  j)r('miiim  ought  a  man  of  Go  to  pay  for  insuring 
Ills  life  for  *7000  for  1  year  ? 

5.  Ten  young  men  of  "25  form  a  clul).  What  is  the  proba- 
l)ility  that  it  will  be  unbroken  by  death  for  ten  years  ? 

6.  The   probability   that  a   planing  mill   will    burn    down 

Avithin  any  one  year  is  .-     AVhat  ought  an  insurance  company 

to  charge  to  insure  it  to  the  amount  of  $3000  for  1  year,  for 
"ii  years,  for  3  years,  and  for  4  years,  respectively  ? 

7.  If  the  probability  that  a  house  will  burn  down  in  any 

one  year  is  ;>,  what  ought  to  be  the  premium  for  insuring  it 

for  s  years  to  the  amount  of  a  dollars? 

Note.  In  casea  like  the  last  two,  it  is  assumed  that  only  one  loss 
will  be  paid  for. 

8.  What  is  the  probability  that  if  a  man  aged  25  marry  a 
wife  of  20,  they  will  live  to  celebrate  their  golden  wedding? 

9.  A  company  insures  the  joint  lives  of  a  husband  aged  70 
and  a  wife  aged  50  for  sJ^SOOO  for  5  years,  the  stii)ulation  being 
that  if  either  of  them  die  within  that  time  the  other  shall  be 
paid  the  money.  What  ought  to  be  the  premium,  no  allow- 
ance being  made  for  interest  ? 

10.  A  man  aged  50  insures  the  life  of  his  wife,  aged  35,  for 

$10,000  for  20  years,  with  the  promise  that  the  money  is  not 

to  be  paid  unless  he  himself  lives  to  the  ago  of  70.     What 

ought  to  be  the  ])romium? 

Note.  In  computations  relating  to  the  inanap:cmcnt  of  11  fe  insurance, 
it  is  always  necessary  to  allow  compound  interest  on  all  paymcmta.  But 
the  above  exercises  are  intended  only  to  illustrate  the  application  of  the 
theory  of  probabilities  to  the  suliject,  and  therefore  no  allowance  for  in- 
terest is  expected  to  be  made  in  tlie  answers. 


BOOK    XI. 


OF  SERIES  AND  Till':  DOCTRIXi:   OP 

LIMITS, 


n 


wn  in  luiv 


CHAPTER    I. 

NATURE     OF     A     SERIES. 

275.  T>et\  A  Series  is  a  succession  of  tcnns  follow- 
iiig  each  otluT  according  to  sonic  general  law. 

Examples.  An  arithmetical  progression  is  u  series  tleier- 
iiiined  by  the  law  that  each  term  shall  be  greater  than  the 
preceding  one  by  th(   same  anion nt. 

A  geometrical  progression  is  a  series  sui^^ocl  to  the  law 
that  the  ratio  of  every  two  consecntive  terms  is  the  same. 

These  two  progressions  are  the  sim[)lest  form  of  series. 

A  series  may  terminate  at  some  term,  or  it  may  continue 
indefinitely. 

Bef.  A  series  which  continues  indefinitely  is  called 
an  Infinite  Series. 

Def.  The  Sum  of  a  series  is  the  algebraic  sum  of 
all  its  terms.  Hence  the  sum  of  an  intinitc;  series  will 
consist  of  the  sum  of  an  intinite  number  of  terms. 

27(5.  The  law  of  a  series  is  generally  such  that  the  n^^ 
term  may  bo  expressed  as  a  function  of  n. 
For  example,  in  the  scries 

1111, 

3+3+4  +  5  +  ^'^- 
1 


the  n^^  term  is 
21 


w  +  1 


••  c 


'I    , 

1 

•I'       ' 


1322 


SI'JRIKH. 


i'  I  A 


M 


,    1 


!  ,■ 


# 


In  the  series      -—  +  -—  +  7^,^  +  etc., 
the  n^^  term  is 


11  {n  +  1) 

D(f.     'I'he  expression  for  the  ??/'^  term  of  a  series  as 
a  function  of  71  is  called  the  General  Term  of  the 

series. 

EXERCISES. 

Express  the  n*'^  term  of  each  of  the  following  series  : 

111^ 
0  •  4       4  •  5        o  •  0 

2.     1-2  +  3.4  +  r).G  +  etc. 

«  (C^  «^  «^ 

4-     ^."^  +  ;3r22  +  4723  +  5V2*  +  ^^^- 

Write  four  terms  of  each  of  tlie  series  having  the  followiiii^' 

general  terms : 

4/^2  _  1 
15.  The  n^'''  term  to  be  t— , -• 

6.  The  i^^'  term  to  be  i  (/  +  1)  (/  +  2)  2;^ 

7.  The  (?i  +  IF  term  to  be  --— — -^.-^ —^^' 

8.  The  (?i  — 1V<  term  to  be  ,  1^   ~  "^     ■ 

277.  The  most  common  nse  of  a  series  is  to  enable  ns  to 

compute,  by  approximation,  the  values  of  expressions  which  ir 

is  difhcult  or  impossible  to  c()m})ute   directly.     Suppose,  fur 

1  4-  X 
example,  that  we  have  to  compute  the  value  of \  when  .'' 

is  a  small  fraction,  say  ^,  and  to  have  the  result  aocnrate  to 

eight  decimals.     We  shall  see  hereafter  that  when  x  is  less  than 
1,  we  have 


CONVERGENCE   OF  ,^E1UE8. 


1  +a: 


=  1  +  J3^  +  Hx^  +  'Zx^  +  etc.,  ad  infinitum. 


Suppose  X  =  -. 


50 


.02. 


We  compute  this  series  thus : 


2  X  .02  = 
Multiplying  by  .02, 


a 


<( 


^  1.02 


.04 

.0008 
.000016 
^0000032 

1.0-1081032 


which  IS  much  more  expeditious  than  dividing  1  02  by  .98. 

It  will  be  seen  that  every  term  we  add  makes  the  tiuotient 
accurate  to  one  or  two  more  decimals,  so  that  there  is  no  limit 
to  the  precision  which  may  be  attained  by  tlie  use  of  the  scries. 

If,  however,  x  had  been  greater  tlian  unity,  the  series  would 

give  no  result,  because  the  terms  2x,  2x^,  2x%  would  have  gone 

on  increasing  indefinitely,  whereas  the  true  value  of  the  frac- 
1  -{-  X 


tiun 


would  have  been  negative. 


1  —  x 
This  example  illustrates  the  following  t  :7o  cases  of  series : 

I.  There  may  he  a  certdin  limit  in  irliich  the  stnu  of 
the  series  shall  approaeli,  as  we  increase  the  ninnher  of 
terms,  but  which  it  can  never  reach,  how  great  soever  the 
number  of  terms  added. 

For  example,  the  series  we  have  just  tried, 

^222 

■^  50  "^  502  "^  503  +  504  +  etc., 

1.02 

approaches  the  limit  j^^j  but  never  absolutely  reaches  it. 

II.  As  we  inerense  the  nanvber  of  terins,  the  sum 
maij  increase  witliout  limit,  or  mail  vibrate  back  and 
]ovth  in  consequence  of  some  terms  being  positive  and 
others  negative. 

Those  two  classes  of  series  are  distinguished  as  converrient 
and  divergent. 


k 


\  i' 


<  <  I  1  , 


I 

I 


n 


'  'r 


m 


324 


SERIES. 


V     < 


'M 


If 


r  I 


Def.  A  Convergent  Series  is  one  of  which  the  sum 
approaches  a  limit  as  the  number  of  terms  is  increased. 

Refer  to  §  213  for  an  example  of  infinite  series  in  geometrical  pro- 
gressions which  have  limits. 

Dif.  A  Divergent  Series  is  one  of  whicli  the  sum 
does  not  approach  a  limit. 

Examples.  The  series  1  +  2-f  3  +  4  +  etc.,  ad  infimUiin, 
IS  divergent,  because  there  is  no  limit  to  the  sum  of  its  terms. 

The  series  1  —  1  +  1  —  1  +  1  —  etc.,  is  divergent,  because 
its  sum  continually  fluctuates  between  +1  and  0. 

Eem.  When  we  consider  only  a  limited  number  of  term?, 
the  question  of  convergence  or  divergence  is  not  important. 
But  when  the  sum  of  tlie  whole  series  to  infinity  is  to  be  cou- 
sidered,  only  convergent  series  can  be  used. 

Notation  of  Sviins, 

278.  The  sum  of  a  series  of  terms  represented  hy 
common  symbols  may  be  expressed  by  the  symbol  i, 
followed  by  one  of  the  terms. 

Example.     The  expression 

means  "the  sum  of  several  terms,  each  represented  by  a." 

AVhen  it  \i^  necessary  to  distinguish  the  different 
terms,  different  accents  or  indices  are  affixed  to  them, 
and  represented  by  some  common  symbol. 

Example.    The  expression 

lat 

means  the  sum  of  several  terms  represented  by  the  symbol  a 
with  indices  attached  ;  that  is,  the  sum  of  several  of  the  (|uaii- 
tities  «j,  flfg,  «3,  ^4,  etc. 

When  the  particular  indices  included  in  the  summa- 
tion are  to  be  expressed,  the  greatest  and  least  of  them 
are  written  above  and  below  the  symbol  2. 


3li  the  sum 


SIG2^   OF  SUMMATION, 


325 


Examples.     The  expression 


i=5 


rn 


.cans:  "Sum  of  all  the  symbols  a,  formed  by  giving  i  '^\  in- 
gral  values  from  i  =.  5  to  i  =  15. "     That  is,  ' 

i=15 

.  i^^m  means  0  +  .>i  +  ^,^  +  3,,,  _^  4,,^  ^  5^^^^ 

^2  (/,i)  means  (1,./)  +  (o,y)  _,_  (3,y)  +  ^^^^y 

.^i  (^y)  =  (,,  2)  +  (,,  3)  +  (/,  4)  +  (i,  5)  +  (/,  6). 

2  ;.!  =  1!  +  3!  +  3!  +  4!  =  1  +  2  +  6  +  24  =  33. 
11 
^li  ^^  7  +  8  +  9  +  10  +  11  =  45. 


J-- 
n-4 

n- 


i=5 


If  =  02  -I-  32  +  42  +  52  =  54. 


EXERCISES. 

Write  out  the  following  summations,  and  compute  their 
vaJucs  when  they  are  purelv  numf^ripnl  • 


I. 

J=7 

2. 

«=6 

^  n  in 

n=l 

4- 

i=8 
i=4 

5. 

I>nk. 

«=4 

7- 

i=4 
1^2 

8. 

n=5 
n=2 

n=6 


6,    "2U  +  i)(y_i). 

n=0 

9.        1 . 

n=o  n  -{-  1 


Express  the  following  sums  by  the  sign  E : 

lo-     ^^o+/^t+//a +7/3+7^4.  II.     13  +  23  +  33  +  43. 

12.     1.3  +  2.3  +  3.4  +  4.5.       13.     1  +  ?  +  ?  +  ^  +  ^. 

/*       o       4       0        6 


i.  f 


i   ^ 


i.     !  J 


■  ( 


«) 

►  u 


'(  !  , 


'  u 


I:  '■"?"?» 


326 


ISliJIUEB. 


If- 


CHAPTER     il. 

DEVELOPMENT    IN    POWERS    OF    A    VARIABLE. 

219.  Among  the  most  common  series  employed  in  math- 
ematics are  those  of  which  the  terms  are  multiplied  by  tiie 
successive  powers  of  some  one  quantity. 

An  example  of  such  a  series  is 

1  +  2.;  +  32;2  +  4:Z^  -|-  hz'^  +  etc., 

in  which  each  coefficient  is  greater  by  unity  than  the  power  of 
z  which  it  multiplies. 

A  geometrical  progression,  it  will  be  remarked,  is  i  series 
of  this  kind,  in  which  the  terms  contain  the  successive  powers 
cf  the  common  ratio. 

The  general  form  of  such  a  series  is 

in  which  tlie  successive  coefficients  «„,  a^,  a^,  etc.,  are  formed 
according  to  some  law,  but  do  not  contain  z. 

Such  a  series  as  this  is  said  to  proceed  according  to  the 
ascending  powers  of  the  variable  z. 

Rem.  The  sum  of  a  series  is  often  equal  to  some  algebraic 
expression  containing  the  variable.  Conversely,  we  may  find  u 
series  the  sum  of  all  the  terms  of  which  shall  be  equal  to  a 
given  expression. 

Def.  A  series  equal  to  a  given  expression  is  cal](  d 
the  Development  of  that  expression. 

To  Develop  an  expression  means  to  find  a  seri(^s 
the  sum  of  all  the  terms  of  which  are  equal  to  the  ex- 
pression. 

The  most  extensively  used  method  of  development  is  tliiit 
of  indeterminate  coefficients. 


INDETEBMINA  TE   COEFFICIENTS. 


327 


IS  a  series 


iient  is  tliiit 


Method  of  Iiicloterniiiiate  Coeffieioiits. 

380.  The  method  of  indeterminate  coefficients  is  based 
iipon  the  following  principles  : 

Let  us  have  two  e(iual  expressions,  each  containing  a  varia- 
ble z,  and  one  or  botli  containing  also  certain  incletennu/aie 
qiianfitieSy  that  is,  quantities  introduced  hy])othetically,  and  not 
given  by  the  original  problem,  the  values  of  which  are  to  be 
subsequently  assigned  so  as  to  fultll  a  certain  condition. 

The  condition  to  be  fultilled  by  the  values  of  the  inde- 
terminate quantities  is  that  the  two  exi)ressions  containing  z 
and  these  quantities  shall  be  made  identically  equal. 

Then,  because  tlie  e([uations  are  to  be  identically  equal,  we 
can  assign  any  values  we  please  to  z,  and  thus  form  as  many 
equations  as  we  please  between  the  indeterminate  quantities. 

If  these  equations  can  be  all  satisfied  by  one  set  of  values  of 
these  quantities,  then  by  assigning  these  values  to  them  in  the 
original  equation,  the  latter  will  be  an  identical  one,  as  re([uired. 

The  student  should  trace  the  above  general  method  in  the  following 
examples  of  its  application. 

381.  Theorem  I.  //  a  series  proeeeclin^  aeenrdiii^ 
to  the  ascending  poicers  of  ci  quantity  is  equal  to  zero  for 
all  values  of  that  quantity,  the  coefficient  of  each  sepa- 
rate term  must  he  zero. 

Proof,  Let  the  several  coefficients  bo  «„,  «i,  Wg?  ^'^■^">  ^^^^ 
z  the  quantity,  so  that  the  series,  put  equal  to  zero,  is 

Because  the  equation  is  true  for  all  values  of  z,  it  must  be 
true  when  z  =  0.     Putting  z  =  0,  it  becomes 

ffo  =  0. 
Dropping  a^,  the  equation  becomes 

a^z  +  ac,z^  +  a^z^  +  etc.  =  0. 
Dividing  by  z,  a^  -f  a^z  +  f^gZ;'^  +  etc.  =  0. 
From  this  we  derive,  by  a  repetition  of  the  same  reasoning, 

flj  =  0. 


I  .1 


1.  I 


\  ^' 


t   • 


I  1    ' 


'I 


!     (:. 


H 


•    I 


i 


.ip 


<i\  • 


328 


SEUlhJS. 


.  » 


.  w 


Continuing  (he  process,  we  find 


a. 


■=  0,    ffg  =  0,    etc.,  indefinitely. 


Theorem  II.  //  two  series  proceeding  by  ascending 
poivers  of  a  (juantity  arc  equal  for  all  values  of  that 
quantity,  the  coefficients  of  the  equal  powers  must  he 
equal. 

Proof.     Let  the  two  equal  series  be 

«(,  +  «i2:  +  «22!^  +  etc.  =  bQ+h^z-^-h^z^  +  Qtc.  (a) 

Transposing  the  second  member  to  tlie  left-hand  side  and 
collecting  the  equal  powers  of  z,  the  equation  becomes 

«o  —  *o  +  («i  —  *i)  ^  +  (^^2  —  ^2)  ^^  +  etc.  =  0. 

Since  this  equation  is  to  be  satisfied  for  all  values  of  z,  the 
coefiBcients  of  the  separate  powers  of  z  must  all  be  zero. 

Hence, 


or 


a. 


a. 


b,  =0, 


Oo  =  be 


etc. 
etc. 


(t^    Oj     0,  Kg    lyg 

0    =    ^0'  ^1    ^=    ^If  "2    —    ^2> 

Exercise.  Let  the  student  demonstrate  these  last  equa- 
tions independently  from  («),  by  supposing  z  =  0,  then  sub- 
tracting from  both  sides  of  (a)  the  quantities  found  to  be  equal ; 
then  dividing  by  z  ;  then  supposing  z  =z  0,  etc. 

Rem.  The  hypothesis  that  (a)  is  satisfied  for  all  values  of 
z  is  equivalent  to  the  supposition  that  it  is  an  identical  equa- 
tion. In  general,  when  we  find  different  expressions  for  the 
same  functions  of  a  variable  quantity,  these  expressions  ought 
to  be  identically  equal,  because  they  are  expected  to  be  true 
for  all  values  of  the  variable. 

Theorem  III.  A  function  of  a  variable  can  only  he 
developed  in  a  single  way  in  ascending  powers  of  the 
variable. 

For  if  we  sliould  have 

Fz^  A^^-  A,z  +  A^z^  +  A^z^  -f  etc., 
and  also      F^  =  B^  -{■  B^z  +  B^z^  +  B^^  -f  etc., 


INDETEUMINA  TE  (JOEFFIC  "^'miS. 


;i29 


these  two  series,  being  eacli  idontically  equal  to  Fz,  must  he 
identically  equal  to  each  other.  But,  by  Tli.  II,  this  cannot  be 
tlie  case  unless  we  have 

Aq  z=z  Bq,    a  I  =  7?j,     Ac,  =  B^,    etc. 

The  coefficients  being  equal,  the  two  series  are  really  one 
and  the  same. 

38'^.  Expansion  by  Indeterminate  Coefficient f<.  The  above 
principle  is  applied  to  the  development  of  functions  in  powers 
of  tlie  variable.  The  method  of  doing  this  will  bo  best  seen 
by  an  example. 

1.  Develop ;  in  powers  of  x. 

Let  us  call  the  coefficients  of  the  powers  of  x  Oq,  a^,  etc. 
Tlie  series  will  be  known  as  soon  as  these  coefficients  are 
known.    Let  us  then  suppose 

=  «Q  -f  a^x  +  a^x'^  +  a^x^  +  etc. 

Here  we  remark  that,  so  far  as  we  have  shown,  this  equa- 
tion is  purely  hypothetical.  We  have  not  proved  that  any 
such  equation  is  possible,  and  the  ([uestion  whether  it  is  j'^ossi- 
ble  must  remain  open  for  the  present.  We  must  find  whether 
we  can  assign  such  values  to  the  indeterminate  coefficients,  a^, 
f/j,  a^,  etc.,  that  the  equation  shall  be  identically  trne. 

Assuming  the  equation  to  be  true,  we  multiply  both  sides 
by  1  +  X.     It  then  becomes 

1  =z  a^  -i-  (a^  -{-  ai)x  +  (r^i  +  a^)  x^  +  etc. ; 
or  transposing  1, 

0  ==  «o  —  1  +(«o  +  ^i)^  +  {a^-\-az)x^  -\-  («2+^3)^  +  etc. 
By  Theorem  I,  the  coefficients  must  be  identically  zero. 
Hence, 

«o  —  1    =0,    which  gives    a^ 
«i  +  «o  =  0, 


a 


a 


«2   +  ^'l    =   0, 
«3   +  ^'S    =   0» 

etc. 


a 


a 


a 


.  =  1; 
«i  ==  —  «o  =  —  1 ; 

^2  =  —  «i  =  1; 

■  -1; 

etc. 


a^  —  —a^ 


f1 


m 


;«() 


SERIES. 


M 


If 


li 


Substituting  tlicse  values  of  the  coefficients  in  the  original 
equation,  it  becomes 

1— x-\-x'^  —  x'^-\-x^  —  etc. 


l-\-x 

This  same  metiiod  can  be  applied  to  the  development  of 
any  rational  fraction  of  which  the  terms  are  entire  functions 
of  some  one  quantity.     Let  us,  for  instance,  suppose 

a  +  hx 


ni  -\-  nx  -\-  px^ 


—  Aq  ■}-  A^x  -\-  A^x^  +  ....+  AnX^, 


Multiplying  by  the  denominator  of  the  fraction,  this  equa- 
tion gives 

a  -{-  bx  =  diAq  -}-  (uAQ-^mAi)  X  -\-  (pAQ-{-,/A^-\-mAQ)x^ 

+  (pA  1  +  nA  2  +  7nA  3)  x^  -f-  etc. 

We  now  see  that  when  i  >  1,  the  coefficient  of  .r*  in  this 
equation  is  rnAi  -f  }iAi_i  -}-  pAi^Z' 

Equating  the  coefficiouts  of  like  powers  of  x, 


mA, 


a 


a,     whence    A^  =  — ; 


711 


mA^  +  71 A  Q  =  h, 
in  A  3  +  nA  ^  4-  pA  0  =  0, 
mA  3  +  w^  2  +  pA  1=0, 


(< 


ti 


i( 


^'. 

b 

m 

— 

-A, 

• 

A, 

E 

m 

^«- 

m    ^ 

A, 

— 

7n 

Ay- 

-  1 
ni'' 

We  have  from  the  general  coefficient  above  written,  when 

*■    >    1»  A      _  ^^       A  P     A 

Ai  =. yii-i Ai-2. 

m  7n 

That  is,  each  cocfficAent  after  the  second  is  the  same 
linear  function  of  the  two  coefficients  next  preceding. 

Such  a  series  is  called  a  Recurring  Series. 

EXERCISES. 

Develop  by  indeterminate  coefficients: 

1  1 

.  2. • 

l—x  1  —  ^x 


I. 


UNDETERMINED    MULTIPLIERS. 


331 


3  fuiictiuus 


3- 

5- 
7- 


1 

— 

X 

1 

+ 

X 

1  +  a; 

1 

+ 

'Ix  -f-  3.7rJ 

1 

2x  4-  3.-:2 

1  +  2a;  +  3x» 


4. 

1  -f-  a; 
1-1' 

6. 

l  —  x 

1   _  O^   _^  .^2 

8. 

1         X 

I  ^  X  —  .r3 


28.*5.  The  development  of  a  rational  fraction  may  also  be 
effected  by  division,  after  the  manner  of  §§  96,  97,  the  opera- 
tion being  carried  forward  to  any  extent. 


Example.     Develop  - 


1  -\-x 


X 


I  -\-  X 
1—x 


1-x 


1  +  "Zx  -\- %x^  + 'Ix^  +  etc. 


'Zx 

2x  —  2x^ 


2x'^-^  0 
2x^  —  2a^ 


2x^,  etc. 


EXERCISES. 


Develop  by  division  the  expressions : 

1  —  2a;  1  +  X 


I. 


1  -\-  X  1  —  X  -{-  x'^ 

284.  Eliviination  hj  Undetermined  Multipliers.  There  is 
an  application  of  the  method  of  undetermined  coefficients  to 
the  problem  of  eliminating  unknown  quantities,  which  merits 
special  attention  on  account  of  its  instructiveness.  Let  any 
system  of  simultaneous  equations  between  three  unknown 
quantities  be 

ax  +    hy  -[-    cz  =1  /?,  (1) 

a'x  +  h'lj  +  c'z  =  h',  (2) 

a"x  +  h"y  +  (^'z  =:  h".  (3) 

Can  we  find  two  such  factors  that,  if  we  multiply  two  of 
the  equations  by  them,  and  add  the  results  to  the  third,  two  of 
the  three  unknown  quantities  shall  be  eliminated  ? 


i 


f1 


'1 


;    I 


K^i' 


:  k 


332 


SERIES. 


Tliis  question  is  answered  in  Iho  following  way: 

If  there  be  such  factors,  let  us  call  thcni  ///  and  71.     If  wo 

multiply  the  first  ecjiUition  by  7n,  tin;  second  by  ;/,  and  add  llie 

product  to  the  third  equation,  we  shall  have 


+  {bm  +  b'n  +  b")  yl=  hn  -\-  h'u  +  //". 
+  {cm  4-  c'n  +  c")  z 


{I' 


In  order  that  the  quantities  y  and  z  may  disappear  from 
this  equation,  we  must  have 


bm  -\-  b'n  -hb"  =  0, 
cm  +  c'n  -\-  c"  =  0. 

Since  we  have  these  two  equations  between  the  quantities 
m  and  n,  we  can  determine  their  values. 
Solving  the  equations,  wo  find: 

DC    —  0   C 


m  = 


n  =  -,— 
be 


bc'~-  b'c  ' 
b"c  -  be" 


b'c 


These  are  the  required  values  of  the  multipliers.  Substi- 
tuting thorn  in  the  equation  {b),  we  find  that  the  coefficients 
of  y  and  z  vanish,  and  that  the  equation  becomes 


'aib'c"  -b"c)  +  a'{b"c-bc") 


X 


-  h{b'c"-b"c')  +  h'(b"c-  be")        „ 

-  be'  -  b'c  "^  ''  ' 

Clearing  of  denominators  and  dividing  by  the  coefficient  of 
X,  we  find 

_  h  jb'c"  -  b"c')  +  h'  {b"c  -  be")  +  h"  {be'  ~  b'c) 
a  {b'c"  -  be')  +  a'  {b"c  -  be")  +  a"  {be'  -  b'c)' 

EXERCISES. 

I.  Find  the  values  of  y  and  z  by  the  above  process  for 
finding  x. 


X 


MULTIPLICATION  OF  SERIES. 


333 


For  this  purpnso  \vv  miiy  ho^xn  with  tlic  equation  (h)  and  r.iul  vnhios 
of  in  and  //  such  that  the  coctlirionts  of  x  and  2  in  (/*)  Mhall  vanish.  Those 
v;iluea  will  he  dlflTercnt  from  those  |,^iv(?n  in  (r).  By  Huijstitutiiifj;  tliem  In 
[bs,  X  and  z  will  be  eliniiiMtod,  and  wo  Hliall  obtain  tho  value  of//. 

Wo  thon  find  a  *air(i  sot  of  valiios  of  m  and  «,  purli  .hat  tho  cocffl- 
cienta  of  x  and  y  onall  vanish,  and  thus  obtain  tin;  valiio  of  2. 

2.  Solve  by  the  motliod  of  imlctrrniiiiate  mr.ltii)liers  tho 
exercise  3  of  §  140. 


Miiltipliciition  of  Two  Infinite  Series. 

284rt,  pROBLKM.     To  express  the  product  of  the  two 

series 


and 


^0  ■+-  ^1^'  4-  f^^'^  4-  «3^  +  etc., 


The  metliod  is  similar  to  that  by  which  the  scjuarc  of  an 
entire  runction  is  formed  (§  173,  'i). 

We  readily  find  the  first  two  terms  of  the  product  to  be 

The  combinations  which  produce  terms  in  0(y^  are 

Those  v/hich  produce  terms  in  1^  arc 

In  general,  to  find  the  terms  in  x^^  we  begin  by  multiplying 
r/fl  into  the  term  hnxV-  of  the  lower  series,  and  then  multiplying 
each  succeeding  of  the  first  series  by  each  preceding  term  of 
the  second,  until  we  end  with  anh^o-^-     Hence,  if  we  suppose 

Product  =  Af^  -{-  A^x  +  A^x^  -\- .  .  .  .  -\-  .4„.i"  +  etc., 

we  shall  have,  for  all  values  of  n, 

An  =  a^bn  +  a^bn-i  +  a^bn-z  +  .  .  .  .  +  «nJo* 

By  giving  nail  integral  values,  we  shall  form  as  many  values 
as  we  choose  of  An,  and  so  as  many  terms  as  we  choose  of  the 

series. 


'    I 


t   I ' 


kt 


■  I 


Mi 

! 

I 

I 


,  1 


'  (  r 


334 


SERIES, 


1 


EXE  RC  I  S  ES. 

1.  Form  the  product  of  the  two  series: 

^       .r*       r^       .«« 

7^         X^         X^  , 

2.  Form  t!)0  square  of  each  of  these  series. 

3.  Can  you,  })y  adding  the  S(|nares  together,  show  that  tlirir 
sum  is  equal  to  unity,  whatever  he  the  value  of  a;? 

To  etTect  this,  imilti|>ly  each  coelllcient  of  x"  in  the  sum  of  the  RquariH 
by  n\,  Hul)8titiit('  for  each  terra  its  value  C"  given  in  J5  257,  and  ai)|)ly 

§\>62/ni.  II. 

285.  Scries  proceeding  according  to  the  Poivers  of  Two 
VariubJes.    Sucli  a  series  is  of  tlie  form 

^0  +  ^o''  +  (^y  +  ^0^^  +  i>x^y  +  ft'ijp  +  etc., 

in  whieh  the  products  of  all  powers  of  x  and  //  are  comhincd. 
By  collecting  the  coefticicnts  of  each  power  of  a;,  the  series  will 
become 

^0  +  «i//  +  f^-jf  +  ^3//^  +  .  .  .  . 
+  (*o  +  '\y  +  h^y"^  +  b:if  4- )^ 

+  (^'0  +  C\y  +  c^y^  +  c^y^  +  — )x'^ 

+  etc.,     etc.,      etc.,      etc. 

Hence,  the  series  is  one  proceeding  according  to  the  powers 
of  one  variable,  in  which  the  coefRcients  arc  themselves  series, 
proceeding  according  to  the  ascending  powers  of  another 
variable. 

Let  us  have  the  identically  equal  series  proceeding  accord- 
ing to  the  ascending  powers  of  the  same  variables, 

/!„  +  A^y^A^'if+ 

+  {B^  +  B^y  +  B^y'^  + )x 

+  etc.,      etc.,       etc. 

Since  these  series  are  to  be  equal  for  all  values  of  x,  the 
coefficients  of  like  powers  of  x  must  be  equal.     Hence, 


SERIES.  335 

«o  4-  a^y  -f  n^if  -f.  etc.  =  A^  +  .1 , y  +  .|gy.>  -|-  i-tc. 
*o  +  ^y  +  h^if  -f  etc.  =  //„  -h  li.if  +  /y^y  4-  etc. 
etc.  etc. 

Again,  since  tlicse  scries  are  to  be  ecimil  for  all  values  of  v, 
wo  must  have 

f^Q  —  ^U»    f(\  =  '1,,    (ffi  =  A„,    etc. 
b^  =  /y^,     b,  =  //j,     /5»8  =  //j,     etc. 
etc.  etc.  etc. 

Hence, //^  order  that  tiro  ficrics  prnrrrdiin^  ar^ni'dincs 
to  the  ascending  powers  of  two  ruriafj/rs  ntrn/  he  identf- 
r((Uij  eqiud,  the  coefficients  of  every  like  product  of  the 
pow'crs  must  be  equal. 


i  ,1 : 


f  ,i; 


'  i 


4    * 

I 

i 


;U) 


i« 


^  ■< 


336 


SERIES. 


1    ' 


r- 


CHAPTER     III. 
SUMMATION      OF     SERIES. 


Of  Figiirate  Numbers. 

286.    The  numbers  in  the  following  columns  are  formed 
acco'-ding  to  these  rules  : 

1.  The  first  column  is  composed  of  the  natural  numbers, 
J.J  /vj  Oj  etc. 

2.  In  every  succeeding  column  each  number  is  the  sum  of 
all  the  numbers  above  it  in  the  column  next  preceding. 

Thus,  in  the  second  column,  the  successive  numbers  are : 

1,     1  +  2  =  3,     1-f  2+3  =  G,     1  +  2  +  3  +  4  =  10,  etc. 
In  the  third  column  we  have 

1,     1+3  =  4,    1  +  3  +  6  =  10,     etc. 


1 

1 

2 

3 

1 

1 

8 

4 

1 

6 

5 

1 

4 

10 

6 

10 

15 

7 

5 

16 

20 

35 

21 

6 

21 

35 

7 

etc. 

etc. 

etc. 

(^) 


It  is  evident  from  the  mode  of  formation  that  each  niiniber 

is  the  difference  of  the  two  numbers  « 

next  above  and  below  it  in  the  col-  «      « 

umn  next  following.  •      •      • 

The  numbers  1,  3,  0,  10,  etc.,  in  •      •      •      • 

the  second  column  are  called  trian-  •      •      o      •      • 

gular  numbers,  because  they  repre-  //=  1+2+3+4+5. 


%  -^ 


SERIES. 


337 


sent  numbers  of  points  which  can  be  regularly  arranged  over 
triangular  surfaces. 

The  numbers  1,  4,  10,  etc.,  in  tlie  third  columns  are  called 
pyramidal  numbers,  because  each  one  is  composed  of  a  sum 
of  triangular  numbers,  which  being  arranged  in  layers  over 
each  other,  will  form  a  triangular  i)yramid. 

All  the  numbers  of  the  scheme  are  called  figurate  num- 
bers. 

The  numbers  in  the  i''^  column  are  called  tigurate  numbers 
of  the  i^^  order. 

287.  If  we  suppose  a  column  of  I's  to  the  left  of  the  first 
column,  and  take  each  line  of  numbers  from  left  to  right  in- 
clined upward,  we  shall  have  the  successive  lines  1,1;  1,  2, 1 ; 
1,  3,3,  1,  etc.  These  numbers  are  formed  by  addition  in  the 
same  way  as  the  binomial  coefficients  in  g  171,  'I.  We  may 
therefore  conclude  that  all  the  numbers  obtained  by  the  pre- 
ceding process  are  binomial  coefficients,  or  combinatory  expres- 
sions.   This  we  shall  now  prove. 


Theorem.     Tlie  n^  nmnher  in  the  i^^  column  is  equal 


to  C 


yn+i-l 


07'  to 


n  (?i  4-  1)  (?i  +  2)  .  .  .  .  (w  +  /  —  1) 


0) 


1 • 2* o  .  .  .  .  t 

Proof.     Because  the  combinations  of  1  in  any  number  are 
equal  to  that  number,  we  have,  when  i  =1, 

71^  number  in  1st  column  =z  n  =  6'", 

which  agrees  with  the  theorem. 

When  i  —  2,  we   have,   by  the   law  of  formation  of   the 
numbers, 
71^^  number  in  2d  column  =  T'l  +  C'l  -f  6'i  +  .  .  .  .  +  T'l, 

« -J- 1 

which,  by  equation  (a)  (§  260,  3),  is  equal  to  ('2    . 

Therefore  the  successive  numbers  in  the  second  column, 
found  by  supposing  n  =  l,  ti  =2,  etc.,  are 


'r3 


ri^i     r'^     r^  f' 


,rni 


92 


I  f 


I  I: 

I  I      ■ 


» '  ; 


If;  '  • 


338 


SERIES. 


>  I. 


r 


Since  tho  ii^^  number  in  the  third  column  is  equal  to  the 
sum  of  all  above  it  in  the  second,  we  have 

71^^^  number  in  3d  column  =  Cl-\-  6'!+  Ct-\-  C'l^  =  Cl'^ 

which  still  corresponds  to  the  theorem,  because,  jvhen  i  =  ;{, 
n  -{-  i  —  \  =z  n  -j-  2. 

To  prove  that  the  theorem  is  true  as  far  as  ^  e  choose  to 
carry  it,  we  must  show  that  if  it  is  true  for  any  value  of  /,  it  is 
also  true  for  a  value  1  greater.  Let  us  then  suppose  that,  in 
the  r^^  column  the  first  n  numbers  are 

/^iV      /-iTil       fit '-2  r^r>rn-\ 

Since  the  n*^  number  in  the  next  column  is  the  sum  of 
these  numbers,  it  will  be  equal  to 

which  is  the  expression  given  by  the  theorem  when  we  suppose 
*  =  r  +  1. 

Now  we  have  proved  the  theorem  true  when  i  :=  3;  there- 
fore (supposing  r  =  3)  it  is  true  for  i  =i  4.  Therefore  (su[)- 
posing  r  =  4)  it  is  true  for  i  =z  5,  and  so  on  indefinitely. 

If  in  the  general  expression  (1)  we  put  i  =  2,  we  shall 
have  the  values  of  the  triangular  numbers  ;  by  putting  i  =  3, 
we  shall  have  the  pyramidal  numbers,  etc.     Therefore, 


The  7?*^  triangular  number 


The  71^  pyramidal  number  = 


^_7i(n-]-l) 


1-2 
n  (71  +  1)  (n  +  2) 

~'    r2.3 


By  supposing  n  =  1,  2,  3,  4,  etc.,  in  succession,  we  find 
the  succession  of  triangular  numbers  to  be 


1:1    I'l  1'^ 

r.2'     1.2'  1-2' 
and  the  pyramidal  numbers, 

1.2-3      2.3.4      3.4.5 


etc. ; 


etc.. 


1.2.3'     1.2-3'     1.2.3' 
which  we  readily  see  correspond  to  the  values  in  the  scheme  (A). 


SERIES. 


339 


wrc  suppose 


Enumeratioii  of  Triangular  Piles  of  Shot. 

288.  An  interesting  applicution  of  the  preceding  theory  is 
that  of  linding  the  number  of  cannon-shot  in  ti  pile.     Tliere 
lire  two  cases  in  which  a  pile  will  con- 
tain a  tigurate  nnmbei  : 

I.  Elongated  projectiles,  in  which 
each  rests  on  two  projectiles  below  it. 

II.  Spherical  projectiles,  each  rest- 
ing on  three  below  it,  and  the  whole 
forming  u  pyramid. 

Case  I.    Elomjated  Projectiles.    Here 
the  vertex  of  a  pile  of  one  vertical  layer  will  be  formed  of  one 
shot,  the  next  layer  below  of  I  wo,  the  third  of  three,  etc. 
Hence  the  sum  of  7i  layers  from  the  vertex  doAvn  will  be  the 
n^^  triangular  number. 

It  is  evident  that  the  number  of  shot  in  the  bottom  row  is 
eijual  to  the  number  of*  rcnvs.  Hence,  if  n  be  this  num' <.'r, 
and  N  the  entire  number  of  shot  in  the  pile,  we  shall  have, 

n{n-\-\) 

If  the  pile  is  incomplete,  in  consequence  of  all  the  layers 
above  a  certain  one  being  absent,  we  lirst  compute  how  many 
there  would  jc  if  the  pile  were  com|)lete,  and  subtract  tiie 
luimber  in  that  part  of  the  pile  which  is  absent. 

Example,  "^rhe  bottom  layer  has  25  shot,  but  there  are 
oidy  11  layers  in  all.     How  many  shot  are  there? 


K  the  pile  were  complete,  the  number  would  be    - 


25-26 


2 


There  being  14  layers  w^anting  from  the  top,  the  total  number 
of  shot  wanting  is    -—  —  •     Hence  the  number  in  the  pile  is 


2 


N- 


25-20-14.15        (14  +  11)  (15  -f  11) -14- 15 


2 


2 


_  11(14  +  15  +  11) 


2 


nz  220. 


♦ '    J ' 


V 


I  ' 


i  ■  \ 


. !: 


340 


,si':itiKs 


! 

1 

i 

,'» 

! 

i  f.'* 


i 


Note.  Tliis  particular  prohlein  could  have  been  solved  more  briefly 
by  considering  the  number  of  shot  in  the  several  layers  as  an  arithmetical 
progression,  but  we  have  preferred  to  api)]y  a  general  method. 

EXERCISES. 

1.  A  pile  of  cylindriciil  shot  has  n  in  its  bottom  Yovi,  and  r 
rows.     How  many  shot  are  there  ? 

2.  From  a  complete  pile  having  h  layers,  s  layers  are  re- 
moved.    How  many  shot  are  left  ^ 

3.  A  pile  has  n  shot  in  its  bottom  row,  and  m  in  its  top 
row.     How  many  rows  and  how  many  sliot  are  there? 

4.  A  pile  has  p  rows  and  k  shot  in  its  top  row.  How  many 
shot  are  there? 

5.  Explain  the  law  of  succession 
of  even  and  odd  numbers  in  the  se- 
ries of  triangular  numbers. 

6.  Uow  many  balls  are  necessary 
to  fill  a  hexagon,  having  ii  balls  in 
each  side  ? 

Note.     In  tlie  adjoining  figure, 

1%  —  3. 

389.  Case  II.  Pyramid  of  Balls.  If  a  course  of  balls 
be  laid  upon  the  ground  so  as  to  fill  an  equilateral  trianali', 
having  n  balls  on  each  side,  a  second  course  can  be  laid  upon 
these  having  n  —  1  balls  on  each  side,  and  so  on  until  we 
ccme  to  a  single  ball  at  the  vertex. 

Commencing  at  the  top,  the  first  course  will  consist  of  I 
ball,  the  next  of  3,  the  third  of  6,  and  so  on  through  the  tri- 
angular numbers.  Because  each  pyramidal  number  is  the 
sum  of  all  the  preceding  triangular  numbers,  the  whole  num- 
ber of  balls  in  the  n  courses  will  be  the  n^^  pyramiaul  number, 
or 

n  (n  4-  1)  {r^  +  2) 


N  = 


1.23 


EXERCISES 


I.  How  many  balls  in  a  triangular  pyramid  having  9  balls 
on  each  side  ? 


tiiatiEs. 


341 


^ers  are  w 


2  If  from  a  Iriangiiljir  jnraniid  of  ;/  courses  k  courses  be 
removed  from  the  top,  how  many  bails  will  l)e  left  ? 

3.  How  many  balls  in  the  frustum  of  a  triangular  pyramid 
luiving  71  balls  on  eacli  side  of  the  base  and  m  on  each  side  of 
the  upper  course? 

Sum  of  the  Similar  Powers  of  an  Arithmetical 

Progression. 

'^90,   Put        ^1,  the  lirst  term  of  the  progression; 
d,  the  common  difference; 
??,  the  number  of  terms; 
m,  the  index  of  the  power. 

It  is  required  to  find  an  expression  for  the  sum, 

(C  +  (rti  +  d)""  +  (^1  +  2r/)'»  +  ....+  [«j  +  (,i  _  1)  ,/]m 

which  sum  we  call  Sm' 

Let  us  put,  for  brevity,  a^,  a^,  a^,  a^,  .  ...  an  for  the  sev- 
eral terms  of  the  progression.    Then 

(tz  =  «i  +  ff> 

flfg  =  rtj  -\-  2d  =:  rtg  -f  d, 

an  =  fli  -i-  («  —  l)d  =  ttn-l  +  d. 

Raising  these  equations  to  lUe  (m  +  l)'^'  power,  and  adding 
the  equation  finn  =  f'n  +  d,  we  have 

f,mn  -  arn^-i  +  {ni  +  ^)fifd  +  ^l±~^^—  a^'-hP  +  etc 
*  1  1  1  •  ^  ^ 

amt  =  a^^^  +  {m  +  l)  a^d  +  ^±-}^—  af-hP  +  etc. 

ay^i  =  a^'-^  -i-  {m  +  l)afd  -+-  i^^ii:^  r^m-1^2  +  etc. 

.  •  •  • 

...  • 

«-;i  :=  ^"'+1  +  {m  -I-  1)  ff^d  +  ^i±-^-li''  a^-kP  +  etc 

If  we  add  these  equations  together,  and  cancel  the  common 
terms,  r/^'M-  ^^ '*  +  ..••  +  a^^^,  which  appear  in  both 
members,  we  shall  have 


i 


I'f* . 


I 
I 


!     I 


■  \ 

1 

' 

f 

i 

■\ 

\ 

''\ 

^42 


SERIES. 


I 


r- 


1 


li 


m 


[m  +  1)  m  (//?  —  1)  „„ 
+  ^— ^iT^Ta -(^^^m-2,  etc. 


From  this  we  ol)taiii,  by  solving  with  respect  to  >s';„, 


S 


fiin  ^  1  fim  1 1 


«  ti 


,-\ 1.2-3   ^--ffl-2— g^^'v  r.') 


wliicli  will  enable  us  to  find  aS^  when  we  know  Si,  iS'g,  .... 
*Sm-i.  that  is,  to  find  the  sum  of  the  n^^  powers  when  we  know 
the  sum  of  all  the  lower  powers.  It  will  be  noted  that  >', 
means  the  sum  of  the  arithmetical  series  itself,  as  found  in 
Book  VII,  Chap.  I  ;  and  that  S^  =  n,  because  there  are  n 
terms  and  the  zero  power  of  each  is  1. 

By  §  209,  Prob.  V, 

To  find  the  sum  of  the  squares,  we  put  m  =  2,  which  gives 


^2  = 


«Li  -  «' 


fP 


3d     --"•''i-3'\- 


(3) 


201.  The  simplest  application  of  this  expression  is  given 
by  the  problem; 

To  find  the  STun  of  the  squares  of  the  first  n  natiir/iJ 
ufCDibcrs,  mnnely, 

r-^  +  33  +  32  +  42  + -f  lA 

Here  d  =  l,  an  =  n,  etc.,  S^  =  1  +  2 -i-n  =  — ^— ^  -   , 

so  that  (3)  gives 

^.    _  (u  +1)3  —  1       71  (n  +  1)       ft 
'  2  -  3  2  3* 

Noting  that  n  +  1  is  a  factor  of  the  second  member,  we 
may  reduce  this  equation  to 

n  {?i  +  1)  {2n  +  1) 


*S'n 


6 


(i) 


"which  is  the  required  expression  for  the  sum  of  the  squares  of 
the  first  /(  numbers. 


8ERIEH, 


:m:3 


/  n  natural 


member,  wc 


2i{yZ.  To  find  the  sum  of  the  ca])es  of  any  progression, 
we  pnt  ir,  =  3  in  the  e(juiition  {2),  whicii  then  gives 


.93  =. 


a 


('!) 


Applying  this  as  before  to  tlie  case  in  whieli  ^,,  n„,  a^, 
etc.,  are  tiie  natural  numbers,  1,  2,  3,  etc.,  we  lind 


c         ''^  +  1)'*  —  1       3  ^         ^ 
*^3  =  1 .)  ^^2  —  '^1 


4-« 


—  (^^  4-1)^  —  1  _  ^<  (w  +  1)  (2y?  +  1)  _  n  {n  +  1)       ?/ 

—         4  ~~  4  a  4* 

Separating  the  factor  n  +  I  and  then  reducing,  this  equa- 
tion becomes 


S.  = 


n  {n  +  1)" 
2 


(5) 


But  — ~T — ^  is  th'^  sum  of  the  natural  numl)ers 


1  +  2  +  3  +  etc., 

and  S^  being  the  sum  of  the  cubes,  we  have  the  remarkable 
relation, 

13  _}_  23  4.  33  + _^  ,^3  ^  (m_  2  -^  3  + +  nf. 

That  is,  the  sum  of  the  euhe.'^  of  the  first  n  numbers  is 
equal  to  the  square  of  their  shiH'. 

We  may  verify  this  relution  to  any  extent,  thus  : 
When  n  =2,  13  +  23=  1  4-8  =  9  =  (l+3)'2. 
When  71  =  S,  P  +  23  +  33=:  1^8  +  27:=  36  =  (1+2  +  3)*. 
When  r?=4,  13  +  23  +  33  +  43  =  1+8  +  27  +  04  =  100  =  (1  +  2  +  3  +  4)'^ 
etc.  etc.  etc.  etc. 

293.  Emimerafion  of  a  Jlecfanf/nlar  Pile  of  Balls.  Tiie 
preceding  theory  may  be  applied  to  the  enumeration  of  a  pile 
of  balls  of  which  the  base  is  rectangular  and  each  ball  rests  on 
four  balls  below  it.  Let  us  put  p,  q,  the  number  of  balls  in 
two  adjacent  sides  of  the  base. 


»' 


i.       ii 


I  J 


^r 


t' 


'HT^ 

1 

1 

'    I 


344 


si'JUIes. 


Then  the  second  course  Avill  have  ;j  —  1  and  q  —  I  balls 
on  its  sides ;  the  third  p  —  2  and  y  —  ^,  and  so  on  to  the  top, 
"vvhieh  will  consist  of  a  single  row  of  p  —  q  -}-  I  balls  (suppos- 
ing p  >  q).  The  bottom  course  will  contain  pq  ball;^,  the  ncM 
course  {p  —  1)  (7  —  1),  etc.  The  total  number  of  balls  in  tlii' 
pile  will  be 
jV  =  pq-{-{p-l){q-\)-^ip-2){q~2)-{-.,..-\-{p-q  +  l).    (f.) 

To  find  the  sum  of  this  series,  k-t  us  first  suppose  p  =  y. 
and  the  base  therefore  a  square.     We  shall  then  have 

N'  ^  r/  +  (r/  -  1)2  +  (y  _  2)2  4-  .  .  .  .  +  1, 

th"  sum  of  the  squares  of  the  first  q  numbers. 

^'V,  ;-y§29i,(4), 

^'  ^Q(Q  4-1)  (^7+  1) 


whicl: 


Tlu 


6 


C) 


Next  let  us  put  ?•  for  the  number  by  which  p  exceeds  q  in 
the  general  exi)ression  (G).     This  expression  will  then  beconu' 

JV=q{q  +  r)  +  {q-l)  {q -l-^r)  +  {q-2)  {q-2-{-r)  +  .  .  .  . 

+  (1  +  /•) 

=   ,/  4.  (^  _  1)2  4-   (y  _  J>)2  +  .  .   .  .    4_  02  +  1 

+  [«?  +  ('7  -  1)  +  ((?  -  ^)  +  •  .  •  •  +  1]  >' 


(§  291,  4.) 


_  y(y  +  i)(2^  +  i)     qJJL±l)  r 

~~  G  "^2 

q{q  +  1)  (3r  +  2*7  +  1) 

G 

EXERCISES. 

1.  Find  the  sum  of  the  first  20  numbers,  l  +  2-f-3+  .  .  .  . 
+  20,  then  the  sum  of  their  squares,  and  the  sum  of  tluii' 
cubes,  by  successive  substitutions  in  the  general  e(iuation  (2). 

2.  Express  the  sum  and  the  sum  of  the  squares  of  the  first 
r  odd  numbers,  namely, 

1  +3   -f-  5  + +  {'2r  —  1), 

and  12  4.  32  _^  52  ^ +  (2r  -  1)2. 

3.  Express  the  sum  of  the  first  r  even  numbers  and  the 
sum  of  their  squares,  namely, 

2   +  4  +  G   + +   2r, 

and  22  +  42  +  62  + +  (2;-)2. 


ShJlilKS. 


n4i"5 


4.  A  rectangular  pile  of  bulls  is  started  with  a  base  of  y; 
balis  on  one  side  and  7  on  the  other.  How  many  balls  will 
there  be  in  the  pile  after  3  courses  have  been  laid  ?  How 
many  after  .s  courses  ? 

5.  Find  the  value  of  the  expression 

2  {a  4-  hx  4-  cx^). 

6.  Find  the  value  of 

2  {a  -^  bx  -\-  cx^). 

394.  To  find  the  sitDV  of  n  terms  of  the  series 

1_         1_         1_  1    _ 

L^  ^  'Z-'d  "^  3-4"^ ^  n{r  +  •)* 

Each  term  of  this  series  maybe  div'  ^d  >  3  two  parts, 
thus : 

j_  _  i_i  JL  _  i_l 

1-2  "12'  2-3    ~  3       3' 

1  1  1 

n 


n  {n  +1)        ti      ?t  +  1 
Therefore  the  sum  of  the  series  is 

(l-2)  +  (3--|)  +  (3-4)+----+Cl-i7Tl)' 

in  which  the  second  i)art  of  every  term  except  the  last  is  can- 
celled by  the  first  part  of  the  term  next  following.  Therefore 
the  sum  of  the  n  terms  is 


1  - 


n 


71  -\-  I  W  +  1 

If  we  suppose  the  number  of  terms  n  to  increase  without 

limit,  the  fraction -r  will  reduce  to  zero,  and  we  shall  have 

n  -\-  1 

r"2  "^  2^  "^  3.1  "^  ^^^*'  "^^  *^{A^"*^2^^i  =  1- 
This  is  the  same  as  the  sum  of  the  geometrical  progression,  ^y  +  a  +  o 

-V  ^  V 


H 


! 


•    '  • 


'  t 


.'11 


•    I 


1  •  -H 


■    , 


i 

■  1 


'f  f  f 


I. 


I 


'If, 


840 


SKIilES. 


+  <'tc.,  nd  iitfinltum.     It  will  be  interesting  to  compare  the  lirst  fow  terms 
of  the  two  aeries.     Tliey  avi'. 

111111 
2      0       12  "^  20      'M      42 

111111 
2      4       H  ^  10  ^  83  ^  04 

We  see  that  the  lirst  term  is  tlie  same  in  botli,  while  the  next  three 
are  larger  in  the  geometrical  jjrogresHion.  Aft<'r  the  fourth  term,  tin- 
terms  of  the  progression  become  the  smaller,  and  continue  so. 

*^il)5.  (renerdUzdlion  of  the  Preccdiufj  liesidf.     Let  us  take 
tho  series  of  which  tlie  w^^  term  is 

V 


(i  +  n-\){j  -\-n-\) 
The  series  to  n  terms  will  then  be 


E 

if 


+ 


+ 


p 


(/  + 1)  ij  + 1)  -^  a  +  ^)-(7  -H  2~) 

+ 


■[        ■    •    •    • 


p 


(/  +  w-l)0-  +  w-l) 
If  we  suppose  ;  >  i,  and  put,  for  brevity, 

^  =J  -  h 
the  terms  may  be  put  into  the  form 

;;  _pa  _  n 

a  ~  k  \i      ir 

p ^    ^ 

J 


(*'+i)(y  +  i) 


-  P I A !__) 


etc. 


etc. 


io 


(*  +  w  ■-l)(y-fw+"l) 


=  P  1-^-1 L— ). 

^•  \t  +  ?2  -—  1     ^  +  ?i  —  1/ 


When  we  add  these  quantities,  the  second  part  of  each  term 
will  be  cancelled  by  the  first  part  of  the  k'^''  term  next  follow- 
ing, leaving  only  the  first  ])art  of  the  first  k  terms  and  the 
second  part  of  the  last  ^-terniii.     Hence  the  sum  will  be 


+ 


k\i      1  +  1 


+  .... 


y+1 


J. 

i-{-n 


i  +  W  — 1' 


J  +  fi 


•i-1/' 


si'jn/Ks. 


347 


5t  few  leriiiH 


Example.    To  find  the  sum  of  h  terms  of  the  series 

^.5  "^  :t.(]  "^4.7  "^  5.«  "^  {n-^  ^){n  -f-  4)* 

Each  term  may  1)0  expri.'.'isiMl  m  \\w  i'urm 

•,].6  ~  ;j\;}      li/' 

---  V^   -') 
4.7"~;i\4     ;/' 

1  ^V.J JLV 

(«  +  l)(w  4-  4)        ;J\;/  4-1       n  -h  4/ 

'rherefore,  separating  the  positive  and  negative  terms,  we 
liiid  tile  sum  of  the  series  to  ha 

l/l       1       1       1  1  1 

•P^  3  4  O  71         91  -\-   I 

_  1    _  1   _  _  1 1 ^1 1 _1       V 

5       G      ""      91      71  +  1      71  +  2      ti  +  'S      ii  +  Aj' 
or,  omitting  tlie  terms  wliich  cancel  each  other, 

1/1       1       1 1 1 1_\ 

3 U  "^  3  "^  4       n'+'}       7i  +  'S       ?i  +  4/' 

When  71  is  infinite,  the  sum  hecomes 

1/1       1       1\  _  1  1=^  _  li^ 
3\2  "^  3  "^  4/  ~  3'12  ~  36' 


EXERCISES. 

What  is  the  sum  of  w  terms  of  tlie  series : 

111, 

I.    --  4-  TT  +  T^.  +  etc. 


3-4   '   4.5    '   5-0 
_1_         1_       J_ 

3.5  ■*"  5-7  "^  7-n 


1  1  1 

2.       :^-^  +  — -   4.    —-  -f 4- 


{2n  4-  l)(2n  +  3) 


1 1 


I  . 


!     I 


\ 

•'  1 

f 

1 

• 

i 

i 
1 

1 

t  ' 
1. 

■    I 

'I 


848 


SKUIKS. 


i- 


3 


3 


3 


3 


^'     TlJ  "^^.-t  "^li-O  ■^••"  '^'n{n  -f-^r 
5.  Sum  tlie  scries 

1  ,  1     __       .   1 J.    f        /  •  / 

«(«  +  1)  "^  (^^  +  1)  («  4-  ^)  "^  (^«  +  Ji)  {a  +  3)  "^     ""•'       '"•^• 

1^9(5.    To  sum  tl»c  series 

.V  =  1  +  '.>;•  +  3?-3  +  4r3  -I-  etc. 

Let  us  first  lind  tlie  sum  ul"  /<  terms,  wliicli  we  slmll  call 
*s;.     Then 

Sn  =  1  +  2/-  -f  '^r^  +  l''^  + nr''-'^' 

Multiplying  by  r,  we  have 

r,Sn  =  r  +  'ir^  +  3/'3  +  4/*'»  -f-  .  .  .  .  +  ?</■«. 
By  subtraction, 

(1  —  r)  Sn  =  I  +  r  +  7-2  +  7"3 +  ?•"-!  —  ///•" 

1  —  r" 


1  — r 

1  —  r^ 


—  nr^  i^'lVl,  Prob.  V). 


Therefore,       *S'„  =  ^  _  ,.)2  ~  1  _  ^ 

Now  suppose  n  to  increase  witliout  limit.  If  ?•  >  1,  tlio 
sum  of  the  series  will  evidently  increase  without  limit. 

If  r  <  1,  both  r"  and  nr'^  will  C(mverge  toward  zero  as  n 
increases  (as  we  shall  show  hereafter),  and  we  shall  have 

V-  — 1- 
"^  -  (1  _  rf 

EXERCISES. 

Find  m  tMie  above  way  the  sum  of  the  following  series  to  n 
terms  and    o  iniinity,  supposing  /•  <  1 : 

1.  a  -\-  '.iar  +  5ar^  +  ^nf^ .  .  .  .  +  (2?i  —  1)  r/r"-^ 

2.  2a  4-  4«/-  +  ikir^  +  8rfr3 .  .  .  .  +  2nar^-'^. 

3.  (r<  +  ^')  r  +  (rt  +  2^)  ;'2  4. -\-  {a  +  nb)  r^. 


HEIUES. 


341) 


..    i 


29t.    Sura  tl>c  series 


_1_  1  1 


1 


(") 


uf  wliich  tlio  L'ciioral  term  is  — , ,,  .         .,- 

^  >/  {n  +  1)  (w  +  2) 

Tit't  us  flud  wliothcr  wo  cuu  express  this  series  as  the  8uni 
( r  two  series.     Assume 


A 


+ 


n 


n  (/<  +  1)  {n  4-  2)       /'  {n  +  1)       (^^  +  1)  {n  +  :e) ' 

nliere,  if  possible,  the  viilues  of  the  indeterniiuate  ('ocflicieuts 

J  jiiul   />  are  to  be  so  chosen  tliat  this  e(iuation  siiall  he  true 

identieally. 

U'educing  the  second  member  to  ii  common  denominator, 

wo  liuvo 

1^ (.1  +  Ii)  n  -I-  2  J 

n  (//  V\)\n  4-  -»)  ~  n  {n  +  'l)"(7/  +  ■>)' 

In  order  tiiat    tliese  fractions  may  l)e  identically  C([ual,  we 

in  list  have 

{A  +  B)  n  -i-  'ZA  =  1,  identicitUy, 

which  recjuires  that  we  have  (§  281), 

A  +  B  =  (t,         2/1  =  1. 


This  gives 

Tlioreforc, 
1 


1 

r 


n  {n  +  1)  {h  -f  2)       2  n  {n  +  1)       2  (?i  +  1 )  (/i  +  2) ' 

so  that  each  term  of  the  series  (a)  may  be  divided  into  two 
terms.     The  whole  series  will  then  be 

\  (r5  +  3^  +  ri  +  '''■)  -  \  (.'^  +  iii  +  4^  +  ''')■ 

We  8(0  on  sight,  that  by  cancellirig  equal  terms,  the  sum  of 


n  terms  is 


Sn  =■  -.  — 


4       2(/i  +  l)(/i  +  2)* 


and  the  sum  to  infinity  is  ^ 


i 


•!■" 


w 


1 11 


350 


SliJlifh'S. 


I         > 


*•      , 


'^1)8.  Consider  tho  h.'innoiiic  series 

111. 


of  wliicli  the  ??/'*  term  is 


1 


n 


'i'liis  series  is  divergent,  because 


we  may  divide  it  into  iin   unlimited  number  of  parts,  each 
equal  to  or  greater  than  ^^ ,  as  follows: 


Ist  term  :=  1,     > 


2d  term  = 


1 


^ 


3d  and  4th  terms  >  J^ ; 


etc. 


etc. 


In  general,  if  we  consider  the  )i  consecutive  terms, 

1 


(a) 


the  smallest  will  be  t--  ,  and  thereibro  Ijieir  sum  will  be  greakr 

than    --  X  fi,  that  is,  greater  than    - 
'-in  ^  '4 

Now  if  in  (a)  we  suppose  n  to  take  tho  successive  values. 
1,  2,  4,  8,  IG,  etc.,  we  shall  divide  the  series  into  an  unlimited 

number  of  parts  of  the  form  {a),  each  greater  than  -•     There- 
fore, the  sum  has  no  limit  and  so  is  divergent. 

Of  DUrereiiees. 

309.  When  wo  have  a  series  of  quantities  proceeding  m - 
cording  to  any  law,  wo  may  take  the  ditferenee  of  every  two 
consecutive  quantities,  and  thus  form  a  scn'ies  of  ditFerenccN 
The  terms  of  this  series  are  called  First  Differences. 

Taking  tho  dillerence  of  every  two  consecutive  diiTcrenco'. 
we  shiill  have  another  series,  the  terms  of  which  are  eailc*! 
Second  Differences. 

The  process  may  be  continued  so  long  as  there  are  any  dil- 
ferences  to  write. 


SEIilES. 


351 


ssive  values. 


Example.     In  the  second  column  of  the  following  table 
are  given  the  seven  values  of  the  expression 

X*  _  lor^  -I-  :]{)x^  —  4(U'  +  '2')  =  (px, 

for  X  =z  Q^  \,  'Z,  3,  4,  5,  0. 

In  the  third  column  a'  are  given  tlic  diU'ereiices, 
G  —  ;>5  =  —  \U,      \  —r>  —  —'),     _  14  —  1  =  —  15,    etc. 

In  column  a"  are  given  the  differences  of  these  dilferenees, 
iiamelv, 


5  -  (-  [!))  ^  +  14, 


1.)  -(-:,)  ^ 


li>,    etc. 


X 

cp.r 

A' 

A" 

A'" 

A'v 

A' 

0 

+  2b 

10 

1 

+  c 

—    5 

+  u 

-24 

2 

+    1 

-15 

—  10 

0 

-\-2-i 

0 

3 

-14 

—  ;>5 

10 

+    24: 

+  24 

0 

4 

—  31) 

11 

+  u 

-\-  48 

-f-24 

6 

—  no 

+  '•^1 

4-  (>2 

6 

+   1 

The  p 

roces.s  is  continued 

to  the 

fourtii 

)r(ler  of 

dilT 

erences, 

which   are   all  (><|ual,  whciiee   llio>;e  ot  the  Urth   and  lollowing 
orders  are  all  zero. 

It  will  he  noted  that  the  sign  of  eacii  difference  is  taiu'ii  so 
that  it  shall  cxjiress  each  <|uantity  )nifiHs  the  quantity  next 
lireeeding.     We  have  therefore  the  following  deiinitions  : 

:$()().  Drf.  The  First  Difference  of  a  functioii  of 
any  variable*  is  tho  increment  of  tin'  function  caused  by 
an  iucrenn'ut  of  unity  in  the  variable. 

The  Second  Difference  is  the  difference  between 
two  consecutive^  first  difterences. 

In  general,  the  i*"'  Difference  is  the  difference  be- 
tween two  consecutive  (/i  —  If  differences. 


I     <  i 


■>rt-^mm 


igiiai 


352 


Sh:i{lE8. 


To  invesli<j;ate  tlic  relation  among  (he  di (Terences,  let  uh 
rei)resent  the  .sueecssive  numhers  in  eaeli  colnmn  l)y  tiie  indiees 
1,  'Z,  .'3,  etc.,  and  let  us  })ut  Aj,  Ao,  Ag,  etc.,  for  the  values  of 
(f>x.  We  shall  then  have  tiie  i'ollo\vin<^  scheme  of  diflerences, 
in  which 

a;   =  a,  —  Ao,      A'j   =  Ag  —  Aj,      a;  =r  A3  -  Ag  ; 

a;  =  a;-a;,    a';=.a;-a;,   a;=:a;-a;; 
^:  =  ^;-^^   a7  =  a:-a';,   a:=.a;-a;; 

etc.  etc.  etc. 

the  n^''-  order  of  differences  Ijeing  represented  by  the  symbiji  A 
with  n  accents. 


a; 


^1 

a; 

1 

Ao 

A', 

M 

^3 

• 

• 

• 

• 

• 

• 

/ 

• 

A,i_i 

^: 


, /// 


A" 


Let  us  now  consider  the  followinp;  problem : 

To  express  Ai  in  terms  of  A^,  a'q,  a'q,  etc. 
We  have,  by  the  mode  of  forming  the  differences, 
A^  =  Ao    f-  a'„,     a'i  ==  a'o  +  Ao,     Aj  =  A'^  +  a'o,  etc.       {<') 

Ag  —  A,  +  Ai,     Ag  =  Ai  +  Aj,     Ag  ==  Ai  +  A      etc. 

If  in  this  last  system  of  equations,  we  substitute  the  vahu  - 
of  A,,  Aj,  etc.,  from  the  system  {a),  we  havp 

A3  rr  Ao  +  '^a'o  -f  a'o,     a;  :=  a',  +  2Ao  +  a'o',  etc.    (/^) 
Again, 
A3  =  A3  f  A'g,     A'3  ==  a'o  +  A'g,     A'3  =  A'a  4-  A'g',  etc. 


D1FFERENCK8. 


353 


or 


Substituting  the  values  of  Ao,  Ao,  etc.,  from  {b),  we  liavc 

A3   ==  Ao  4-  'M'o  +     a'o 

+  a;  +  ^a;  +  a; 

A3  :=  Ao  +  ."iA;  +  15a;  +  a';  (r) 


A3  =  A„  +  2a„  +    A, 

:f  a'o  +2a;+a'o^ 


^0 


A'3    rr:   a'o   +  JJA^   +  IJA^  +  A'J 

Forming  A4  =  A3  +  A3,  etc.,  we  see  that  the  coefficients 
of  Ao,  a'o,  etc.,  which  we  add,  are  the  same  as  the  coefficients 
of  Mie  successive  powers  of  x  in  raising  1  -f  x  to  the  n^f'  power 
•/  successive  multiplication,  as  in  §  171.  That  is,  to  form  A^, 
A'^,  etc.,  the  coefficients  to  be  added  are 

13     3     1 
1_  3     3     1 

1     4     G     4     1 

and  these  are  to  be  added  in  the  same  way  to  form  Ag,  and  so 
on  indelinitely.  Hence  we  conclude  that  if  i  be  any  index,  the 
law  will  bo  the  same  as  in  tlie  binomial  theorem,  namely. 


A;  ==  Ao  +  /a;  +  Q  a;  +  (!,)  a'o'  +  etc.  1 
a;  ^  A'o  -f  ^'A'o  +  (!,)  A';  +  ('3)  A'o'  +  etc.  ) 


i^f) 


To  show  rigorously  that  this  result  is  true  for  all  values  of 
/,  we  have  to  prove  that  if  true  for  any  one  valu(»,  it  must  be 
true  for  a  value  one  greater.  Now  we  have,  by  definition, 
whatever  be  i, 

Ai^i  =  At  +  Ai,         Ai-i  =  Aj  -}-  Ai,      etc. 
Hence,  substituting  the  above  value  of  Aj  and  Aj, 

!.  Ai.i  =  Ao  +  (/  +  i)a;  + 1  (^)  +  'J a; 

r 


+  '_Q+(;)J^o4-etc.         (.) 


23 


If. ,        ' 


354: 


SERIES. 


We  readily  j  rove  tliaL 


etc.  etc. 

Substitufiiii^  these  values  in  (e),  the  result  is  tlic  same  given 
!>y  the  e(|uati()ii  {(/)  when  we  put  /  +  1   lor  /. 

'I'lie  form  (r)  sliows  tiie  tbrmuhi  to  be  true  for  i  =  3. 

'Therefore  it  is  true  I'or  /  =  4. 

Tlieretbre  it  is  true  for  /  =  o,  etc.,  indefinitely. 

EXAMPLES    AND     EXERCISES. 

I.  Ilavino-  rriven  A^  =  7,  A^  —  5,  ill  —  —  2.  and  A'",  A'\ 
etc.  =  0,  it  is  rerinirod  to  find  the  values  of  Aj,  Ao,  A3,  etc., 
indefinitely,  both  by  direct  computation  and  by  the  formula  {>/). 

We  start  the  work  thus: 

The  nuinlxTs  in  coliuun  A"  are  all 
oqiial  to  -  2.  IxTuus,'  A'"  =  0. 

Kach  munb(>r  in  coluinn  A'  after 
the  first  is  found  by  adding  A"  or  —  3 
to  the  one  next  alxive  it. 

Eacli  vahie  of  Af  is  then  obtained 
In 'III  tlie  one  next  al)ove  it  by  addin;.^ 
the  apj^ropriate  value  o*"  A^ . 

This  jM-ocesH  of  ;  ddJticv  -"dn  be 
curried  to  any  exti'nt.  Continuing  it 
to  i:=  10,  we  tshu';  liud   A,,,    .  --'A'o. 

Next,  the  nronera!   formula  y^:)   nrivcs,  ])y  putting  A^ 
A'^  —  5,  A'^^  =  —  5i,  and  all  following  values  —  0, 

2       ' 


I 

Ai 

^i 

A. 

0 

7 

+  5 

1 

4-  VI 

+  3 

-2 

5i 

4   15 

+  1 

«. 

ote, 

1 

-2 

4 

—  1 
etc. 

-^ 

etc. 

i^iV. 

Ai  =  7  +  5/  — 


and  the  student  is  now  to  show  that  by  putting  i  ==1,  /  —  -i, 
etc.,  in  this  expression,  we  obtain  the  same  values  of  Aj,  A... 
A3,  ....  Aio,  that  we  get  by  addition  in  tlic  above  scheme. 

It  is  moreover  to  be  renuirked  tluit  wi;  can  reduce  the  last 
equation  to  an  entire  function  of/,  thus: 

Ai  =  7  +  Gi  -  r'. 


DTFFKRENCES. 


355 


nd 

A'",  A", 

\.y,    A3,     ('tC, 

oniuihi  {(l). 

Ai 

ft 

A,. 

-5 

0 

1    *> 
-  0 

-2 

h  1 

0 

-  I 

-2 

>lr. 

etc. 

i»g 

Ao  -=  7 

2.  Ilavinn^  given  A^  =  5,  A'^  =  —20,  A'^  =  —'M), 
A'^'  =  +  '•,  it  is  reqi'irod  to  find  in  the  sanu'  wuv  tlio  values 
of  A,  to  Aq,  Jind  to  express  Ai  as  an  entire  function  of  i  by 
lorniula  {(/). 

3.  On  March  1,  1881,  at  Greenwich  noon,  the  sun's  lonj^ji- 
tii(U'  was  :Ul°  5'  10  '.!)  ;  on  March  2  it  was  *,n-eater  liy  1'  0'  !)  '.G, 
hut  tlii.s  (hiiiy  increase  was  (liniinishiui;  by  2"  eacli  (hiv.  It  is 
required  to  coni|)ute  the  longitude  for  tiu'  first  seven  days  of 
tlu'  nu)nth,  and  to  llnd  an  expression  for  its  value  on  the  11^^ 
day  of  March. 

4.  A  family  had  a  reservoir  containing,  on  the  morning  of 
May  5,  405  gallons  of  water,  to  which  the  city  added  regularly 
50  gallons  per  day.  The  family  used  'S'y  gallons  on  May  5, 
and  o  gallons  more  each  sul)se(|uent  day  than  it  did  on  the  day 
preceding.  Find  a  general  e.xi)ressioM  for  tlu;  (puuitity  of 
water  on  the  /t'^  day  of  May  ;  and  by  e<|uating  this  ex})ression 
to  zero,  find  at  what  time  the  water  will  all  be  gone.  Also  ex- 
plain the  two  answers  given  l)y  the  e([uation. 

Theoroiiis  of  Differonces. 

301.  To  investigate  the  general  properties  of  ditTorences, 
v,e  use  a  notation  slightly  different  from  that  just  employed 

If  u  be  any  function  of  x,  which  we  may  call  <pXj  so  that 
we  put 

ihen  A?<  —  0  (.r  -f  1)  —  0j.  (•/) 

Here  the  symbol  A  does  not  repres*  a  multiplier,  but 
merely  the  words  difference  of. 

The  second  difference  of  v  being  the    'ifference  of  the  dif- 

I'eivnce-  may  be  represented  by  AA?r. 

For  brevity,  ue  put 

A^//   for  AA//. 

where  tb    index  2  is  not  an  exponent,  but  a  symbol  ifsdicating 
;i  second  difference. 

Continuing  the  same  notation,  the  ?/''*  difference  will  be 
re})resented  by  A". 


f'l 


356 


SERIES. 


f 


EXAMPLE. 

To  find  tlie  siiccossivo  clifTLTonci's  of  the  function 

u  =  n.r^  +  bx\ 
By  the  iorninla  (a),  \\v  have 

An  =  a  (./•  -f  1)3  ^-  /y  (r  -f-  1  )2  _  ^rx^  —  hx^  ; 
and,  by  dc'V('k)|)iiig, 

Ml  =  :](ix^-  +  (:3r^  +  2b)  .r  -{-  a  -\-  b. 
Taking  tlie  difference  of  this  last  e(juation, 

Aht  =  3a  [x  +  ly  -{-  {3a  +  )ib)  (.r  -f-  1)  +  a  +  b 

—  3ax^  —  {'3a  +  2b)  x  —  a  -  b 
—  Gnx  -I-  C)a  -f  2b. 

Again  taking  the  difference,  we  liave 

A^u  —  Cm  (:r  +  1)  —  (Ufx  =  On. 
This  expression  not  containing  x,  A*u,  A^w,  etc.,  all  vanisli. 


EXERCISES 


Compute  the  differences  of  the  functions 


I.     .r  +  tnx^  -\-  )ix  -]-  ]). 
3.     5./-3  -\-  10^'^  -f  15. 


2.     2x*  -^  3.^2  ^  5. 


4.  In  the  case  of  the  fast  expression,  prove  the  agreement 
of  results  by  comjMiting  the  values  of  An,  A'^ti,  etc.,  frx-  x  =  0, 
x  =  I,  and  ;r  =r:  ;},  and  comparing  them  with  those  obtaini^d 
by  the  metiiod  of  §  209.  The  latter  are  shown  in  the  follow- 
in  sr  tablo: 


X 

0 


u  =  5a^+  10.^2  -1-  15. 


ti 


15 


30 


95 


240 


A2l 


15 


65 


145 


255 


Ahi 


50 


80 


110 


A^u 


30 


ao 


BrFFf'JIih'M'ES. 


:r)7 


5.  Do  the  same  thing  fur  exercise  2,  luul  for  the  fiinclioii 
tahii lilted  in  §  'VM. 

myZ,  It  will  he  seen  hy  the  ])receding  exiimples  aiid  exer- 
cises, that  for  each  dilfereiice  of  an  entire  function  of  ./•  whitdi 
we  form,  the  degree  of  the  function  is  diniinislieil  hy  unity. 
This  result  i.s  generalized  in  the  following  theorem: 

T/ir  n^  dijfrreiiceH  of  the  J'linction  a*"  ai'e  constcuit 
and  cqiud  to  n  ! 

Proof.  If  u  =  a-'\  we  have,  hy  the  definition  of  the  sym- 
bol A, 

A«  =  (^  +  J )"  —  .r", 


or 


C^ii  =  n.r»   »  +  (  J.r"-'^  +  etc. 


0' 


That  is,  ////  takin<J  the  di/fcrcnrc,  the  highest  po/rrr  of 
x  is  luuUiplied  hi/  its  eA'poucnt  (did  the  fatter  is  diitiin- 
ishect  hi/  luiity. 

Continuing  the  process,  we  sliall  lind   the  u"^  diil'erenco 

to  be 

n{)i  -  !)(//  -'I) 1  =  //! 

Cur.     If  we  have  an  entire  function  of  x  of  the  degree  n, 

ax'^  +  />.r«  »  +  r.r"  *■*  -f  etc., 

the  {ii  —  iy  diiTerence  of  hx»  K  the  {n  —  2)'^  difference  of 
rx"''^  etc.,  will  all  be  constant,  and  therefore  the  n^^^  dilference 
of  these  terms  will  all  vanish.  Therefore,  the  n^^  dilference  of 
the  entire  function  will  be  the  same  as  the  u^^  difference  of 
ax'^ ;  that  is,  we  have 

A"  (ff.2"  4-  *^""^  +  etc.)  =  nji ! 

Hence,  the  n^'-  di/fer'ence  of  a  function  of  the  n^^  de- 
cree is  constant,  and  equal  to  u\  malti plied  hy  the  coeffi- 
cient  of  the  highest  power  of  the  variable. 


»' 


358 


SERIES. 


CHAPTER    IV. 
THE     DOCTRINE     OF     LIMITS, 


\\ 


IM)^.  The  doctrine  nf  limits  oml)ni(!C.s  a  set  of  principles 
.'ipplicablo  to  cases  in  wliieii  the  usual  nietliods  of  calculation 
fail,  in  coiiseciuence  of  some  of  \\\v  (piantities  to  be  used  van- 
ishing or  inereiusing  without  limit. 

We  have  already  made  extensive  use  of  some  of  (he  ]>rinei- 
pies  (»r  I  his  (h)etrine,  and  Ihus  familiarized  the  student  with 
their  ap[>lieation,  but  our  further  advance  requires  that  they 
should  be  rigorously  develojx'd. 

^\.\i()M  I.     \\\y  (iiiMiitity,  however  small,   may  Ix* 
multiplied  so  often  as  to  exceed  any  other  iixed  quau 
tity,  however  g;i'oat. 

Ax.  II.  CoitremcJi/^  any  (jiiantity,  however  pT(»at, 
may  be  dividfnl  into  so  many  ])arts  that  each  juirt  shall 
be  h'ss  than  any^  other  fixed  ([uantity%  howev(^r  small. 

Dcf.  An  Independent  Variable  is  a.  quantity  to 
wiii(di  we  may  assign  any  value  we  please,  however 
small  or  great. 

Theorem  I.  If  a  fraction  have  any  finite  nunirrator, 
and  an,  independent  rariable  for  its  denominator,  we 
mail  assign  to  this  denoTninator  a  i-aUtr  so  £reat  that 
the  fraction  shaft  tie  less  than  any  quantity,  hoivevcr 
sm(dl,  irhich  we  may  assi^'n. 

Proof.  Let  a  be  the  numerator  of  the  fraction,  x  its  de- 
nominator, and  tc  any  (juantity,  however  small,  which  we  may 
chooso  to  assign. 

Let  n  be  the  namber  of  times  we  must  multiply  «  to  make 
it  greater  than  a.     (Axiom  I.)     We  shall  then  have 

a  <  na. 
a 


rottijseenentlv, 


n 


<  «. 


LIMITS. 


359 


ITonce,  by  taking  x  greater  tlmn  w,  we  sliiill  have 


(I 


<  «. 


Example.     Let  a  =  10.     Tlicn  it"  wo  take  for  «  in  succes- 


1 


777-,  ete.,  we  liave  only  to  take 


100     10,000      1,000,000 

X  >  1,000,     x  >  100,000,     X  >  10,000,000,     etc. 


i(»  make         less  than 

X 


«. 


In  the  language  of  limits,  the  above  theorem  is  expressed 


liiis 


(/ 


T/ic  /i III  it  of     ,   if/irn  x  is  iiuLcjinitcl ij  incrcdscd,  is 


zero. 


'  t 


vmrrntor. 


Thkohrm  it.  Ifdfnirtion  havcany  fiuiti'  innnrrfitor, 
and  an  iftdr/wNf/cnt  raridhir  for  i/s  dcnoiniiuitor,  trn 
linn/  ((ssio/i  to  fin's  dcnoni iuffto/'  (t  raliic  so  sinnil  tliot 
t/ic,  fraction,  s/ifd/  exceed  atifi  (/itafititij,  lioirerci'  grcjit, 
I'-hieh  ice  mat/  assign. 


Proof.     Pnt  as  Ix'fore      for  tlie  fraction,  and  let  A  be  any 

number  however  great,  wliieh  we  choose  to  assign. 

Let  n  be  a  number  greater  than  .1.     Divide  a  into  n  parts, 
and  let  «  be  one  of  tiiesc  parts  ;  then 


a  =  im. 


('onse((uently, 


a 


=1  n. 


iC 


Tiierefore,  if  we  take  for;f  a  ([uantity  less  than  «,  we  shall 


ave 


a 


-  >  n  >  A, 


X 


or 


a 


-  >  A. 


X 


Rem.     If  we  have  two  invlependent  variables,  x  and  y\ 
We  may  make  x  any  number  of  times  greater  than  y. 


i  I 

t  ! 


:\(]o 


/j.\f/rs. 


«iir 


i, 


TIr'h  wo  miiy  make  ?/  any  minibor  of  times  preater  tliaii 
this  valiif  «»r  ./•. 

'riiiii  we  may  make  ./•  any  niimlMT  <>f  limes  greater  than 
this  value  i>\'  i/. 

And  wi-  can  thus  continue,  mai\injj:  eacii  varial)Ie  outstrip 
the  other  to  any  extent  in  a  I'ace  toward  inllnity,  wilhout 
either  ever  reaching  the  goal. 

'I'lllioiiKM  III.  //'  /■  /k'  <nni  /i.vcfj  qmiii/ih/.  hoirrrn 
great,  itud  «  a  (jttdntii ij  ii'hicli  ire  nmij  intikc  <is  snmll 
(IS  ICC  jilcnsc,  ICC  nun/  nutk'c  the  prod  net  kn  less  thdnduij 
(issijiiiahic  (/iiftiititi/. 

Proof.  If  tliere  is  any  smallest  value  «)t' /vc,  let  it  be  a. 
Because  we  may  make  re  as  small  as  we  please,  let  us  ])ut 

« <  I' 

^lultiplying  by  k,  we  llnd 

k(c  <  s. 

So  that  ktt  may  ])o  made  less  than  s,  and  .s'  cannot  be  ilic 
smallest  value. 

Dcf.  Tlie  Limit  of  a  vnriahlo  (iiiantity  is  a  value 
wliich  it  can  never  reacli,  l)ut  to  which  it  may  a])])i-()ac|i 
so  nearly  that  tlui  dill'erence  shall  be  less  than  any 
assignable  cjuantity. 

Hem.  In  order  that  a  variable  J^  may  have  a  limit,  it  must 
be  a  function  of  some  other  variable,  and  there  must  ho  cert;iiii 
values  of  this  other  variable  for  which  the  value  of  X  eannni 
be  directly  computed. 

EXAMPLES. 


I.  The  value  of  the  expression 


Y  __  ofi  ~  a^ 
'"  X  —  n 


can  be  computed  directly  for  any  pair  of  numerical  values  of  x 
and  a,  except  those  values  which  are  equal.  If  we  supiwiso 
X  =  a,  the  expression  becomes 


o.. 


LIMITS. 


:mi 


0»  — ff"  _  0 

(t  —  a   —  0' 
which,  eoiisidcivd  hy  ilsclf,  iius  no  meaning. 

2.  Thu  sum  of  any  linitr  nimiluT  <i|'  tt  rnis  uf  a  pcotiu'trioal 
jiro;rrt'8sion  may  be  cuinputrd  hy  n<l<lin<,'  Ihi'in.  lint  if  thr 
iiuniluM-  nt'tirnis  is  intinite.  an  intinilc  time  would  l>e  r(M|uiivd 
lor  the  direct  calculation,  which  is  thefelon'  inipossildc. 

3.  The  area  of  a  |)<»ly;j;on  of  any  nnmher  of  sides,  and  hav- 
ing a  given  aitothegm,  may  he  romputed.  Mut  if  the  nnmher 
of  sides  hccomt's  inllnite,  and  (he  polygon  is  thus  changed  into 
a  circle,  the  ilirect  computation  is  not  practicable. 


EXERCISE, 


Tf  we  have  ihe  fraction,   A' =  ,'         ,  .  show  thai  we  maf 

dr  —  1 

7  1 

make  r  so  great  that  J'  shall  dilfer  from  .,  by  less  than  -r  — » 
^  3     -^  KM) 


''*»  "'""  Tol.^noo'  '^'''''  "'»"  MMiiooi, 


,  and  so  on  inddinitely. 


Notation  of  the  Method  of  LiniitH. 

iU)l.  Put  X,  tlie  (pnintity  of  which   the  value   is  to  be 

foumi  ; 

Xy  the  indcj)end(Mit  variable  mu  which  .V  de- 
pends, so  that  A' is  a  function  of./-; 

a,  the  particular  value  of  x  for  which  we  can- 
not compute  X; 

L,  the  limit  of  A\  or  the  value  to  which  it 
ap})roaches  as  ./•  approaches  to  n. 

Then  the  limit  L  must  be  a  quantity  fultilling  these  two 
conditions : 

1st.  Supposing  X  to  approach  as  near  as  we  please  to  a,  we 
must  always  l)e  fible  to  tind  a  value  of  x  so  near  to  a  that  the 
difference  L  —  X  shall  become  less  than  any  assignalde  quan- 
tity. 

M.  X  must  not  become  absolutely  equal  to  />,  however 
near  x  mav  be  to  n. 


i     t 


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inilTS. 


f.^- 


Rem.     The  qnantity  a,  to^vard  wliicli  i>-  approaches,  may  be 
either  zero,  infinity,  cr  some  liiiitc  quantity. 

Example  i.     Suppose 


A  zz:  

A'  —  a 


I3y  §  93,  this,  expression  is  equal  to 

except  when  a;  =  rt.     But  suj^pose  'S  to  be  the  ditlerence  be- 
tween X  and  a,  so  that 

Substituting  this  value  iu  the  expression  (a),  the  equation 
becomes 

'vS n^ 

X  —  ^f 

Now  we  may  suppose  6  ^o  ymall  that  3«f5  -[-  (52  shall  be  less 
than  any  quantity  we  choose  to  assign.     Hence  we  may  choose 


a  value  of  x  so  near  to  «  that  t  lie  value  of 


x^  —  a^ 
X  —  a 


shall  differ 


from  ?)d'^  by  less  than  any  asrii^tial>]e  cirsantity.     Hence,  if 


■ji.^  —  a 


then 


X  —  a 


r' 


aP 


or  3«'^  is  the  limit  of  the  expiepsion  -^ —  as  x  approaches  a. 

*v     (.V 

X 

Ex.  2.  The  limit  of  -_-,  wbon  x  becomec  indefinitely 

great,  is  unity. 

For,  subtracting  this  expression  from  unity,  we  find  the 
difference  to  be 

By  taking  x  suflBciently  grea*,,  we  may  make  this  expression 

less  than  any  assignable  quant'ty.     (§  303,  Th.  I.)     Therefore, 

x 
approaches  to  unity  a-j  x  increases,  whence  unity  is  its 

X  -|"  1 

limit. 


W 


LIMITS. 


Xotalion.    The  statement  tluit  L  is  the  limit  of  X  as  x 
approaches  a  is  expressed  in  the  form 

The   conclusions  of   the  last   two   examples   may  be  ex- 
pressed thus : 


Lim. 

X  - 


a 


{r=a) 


3rt2. 


X 


Lim.    --- ,  C'-^x)  z=z  1. 
X  4-  1 


l^KM.  This  form  of  notation  is  often  used  for  the  follow- 
ing purpose.  Having  a  function  of  x  wliich  we  may  call  X, 
the  form  X{x=^a)  means,  ''  the  value  of  X  when  x  =  «." 


0. 


EXAMPLES. 

(x^  +  a)(3'=„)  =  a^  +  fi.  (./2  —  a^)^j'^a) 

If  we  require  the  limit  of  a  fraction  wlien  both  terms  be- 
come zero  or  infinite,  divhlc  hotli  terms  hij  some  common 
factor  which  becomes  zero  or  in.finitf/. 

Rem.  If  the  beginner  has  any  difRrnlty  in  nnderstanding  the  pre- 
ceding exposition,  it  will  bi  mllicient  for  him  to  think  of  the  limit  as 
simply  the  value  of  the  exi)ression  when  the  (fuantity  on  which  it  de- 
l>pnds  becomes  zero  or  infinity. 

J' 


For  instance, 


Lim.      "     .     (.r  =  oo), 
.»■  +  1 


tlic  value  of  which  we  have  found  to  be  unity,  may  be  regarded  as  simply 
tli(^  value  of  the  expression,  oo 

x>  +  1 

Although  this  way  of  thinking  is  convenient,  and  generally  leads  to 
C'liToct  results,  it  is  not  mathematically  rigorous,  because  neither  wro 
nor  infinity  are,  properly  speaking,  mathematical  quantities,  and  people 
arc  often  led  into  paradoxes  by  treating  them  as  such. 


Find  the  limit  of 

X  —  a 


EXERCISES. 


X 


when  X  approaches  infinity. 


2. 


Divide  both  terms  by  x. 

V when  X  approaches  infinity. 


mx^ 


px^ 


ax 


when  X  approaches  infinity. 


P 


:504 


LIMITS. 


4. 

5- 
6. 


X 


1  —  ax 


when  X  approaches  infinity. 


X^  —  «2 


X  —  a 

a  +  2; 
a  —  .r 


-  when  X  approaches  a. 
when  .T  approaches  infinity. 


Properties  of  Limits. 

SOili.  Tni:oiiEM  I.  //  ^/fo  functions  are  equal,  thcij 
must  Jia^ve  the  same  limit. 

Proof.  If  possible,  let  L  and  L'  be  two  different  limits  for 
the  rcspL'otive  functions.     Put 

z  =  \{L-L'), 

so  that  L  aiid  V  differ  by  '^z. 

Because  Z  is  the  limit  of  the  one  function,  the  latter  may 
approacli  this  iimit  so  nearly  as  to  differ  from  it  by  less  than  z. 

In  the  same  way,  the  other  function  may  differ  from  L' 
by  less  than  z.  Then,  because  L  and  L'  differ  by  2^:,  the  func- 
tions would  differ,  which  is  contrary  to  the  hypothesis. 

Theorem  II.  The  limit  of  the  sum  of  several  func- 
tions is  equal  to  the  sum  of  their  separate  limits. 

Proof.     Let  the  functions  be  X,  X',  X",  etc. 
Let  their  limits  be     L,   L',  L'\  etc. 

Let  their  differences  from  their  limits  be  «,  a,  a',  etc. 

Then  X  =  L  -  a, 

X'  =  IJ  -  «', 

=    Xy      —  «    , 

etc.  etc. 

Adding,  we  have 

X4-.r  +  .r"  +  etc.  =  Z  +  Z' -I- Z"  + etc. -(«  +  «'  +  «"  + etc.) 

The  theorem  asserts  that  we  may  take  the  functions  so  near 
their  limits  that  the  sums  of  the  differences  «  +  «' -f  ec" -|- etc. 
shall  be  less  than  any  quantity  we  can  assign. 


LfMITS. 


3G5 


Let    k  be  this  qnanUfv,  wliicli  may  be  ever  so  small ; 
n,  tlie  number  of  the  quantities  «,  a',  a",  etc.  ; 
«,  the  largest  of  tlicm. 

Because  we  can  bring  the  functions  as  near  their  limits  as 
we  please,  we  may  bring  them  so  near  as  to  mjike 


or 


na  <  I: 


Then     «-}-«'  + fc"  +  etc.  <  na  (because  ^c  is  the  largest) ; 

whence,  «  +  «'4-«"  +  etc.  <  k. 

Thereforo  the  sum  X-\-X' -{-X" -\-ctc.  Avill  approach  to 
(lie  sum  L  -{-  L'  -\-  L"  -f  etc.,  so  as  to  cHlTer  from  it  by  less 
than  k.  Because  this  ((uantity  k  may  be  as  small  as  we  please, 
A  -{-L'  +  L"  +  etc.  is  the  limit  of  x\ X'  +  X"  +  etc. 

Theorem  III.  TJic  limit  of  the  ]}rodiLct  of  two  func- 
tions is  equal  to  the  product  of  their  limits. 

Proof.     Adopting  the  same  notation  as  in  Th.  II,  we  shall 

have 

XX'  =  LL'  -  aL'  -  (c'L  +  ««'. 

Because  L  and  L'  are  finite  quantities,  we  may  take  «  and 
k' so  small  that  aL'  +  a'L — aa  shall  be  less  than  any  quan- 
tity we  can  assign.  Hence  XX'  may  approach  as  near  as  we 
please  to  LL',  whence  the  latter  is  its  limit. 

Cor.  1.  Tlie  limit  of  the  prod  net  of  any  niimhcr  of 
functions  is  equal  to  the  product  of  their  limits. 

Cor.  2.  Tlie  limit  of  any  power  of  ci  function  is  equal 
to  the  power  of  its  limit. 

Theorem  IV.  TJie  limit  of  the  quotient  of  tiro  fnne- 
fions  is  equal  to  the  qitotient  of  their  limits. 

Proof.  Using  the  same  notation  as  before,  we  have  for  the 
quotient  of  the  functions, 

X'       L'  —  « 

'  -  -  -^-       - 


X 


(C 


V 


while  the  quotient  of  their  limits  is  y 


•r 


'  % 


4m 


fS  ■■ 


306 


LIMITS. 


TIiu  diilerencc  between  the  two  quotients  is 

L'       IJ  —  fc'         ha  —  X  a 
L        L  —  a         L{L  —  «) 

If  L  is  different  from  zero,  wc  nuiy  make  the  quantities  « 
and  a  so  small  that  this  expression  shall  be  less  than  any 

(puintity  we  choose  to  assign.     Therefore,  y  is  the  limit  of 

-' ,  that  is,  of  ^« 

■f^n fiTi 

3<)(>,  Problem.     To  find  the  limit  of  ' as  x 

'  X  ct 

approdches  a. 

Case  I.     When  n  is  a  positive  lohole  number. 
We  have  from  §  93,  when  x  is  different  from  a, 

rfTh    flTl 

X  —  a 

Now  suppose  X  to  approach  the  limit  a.  Then  .t"~^  will 
approach  the  limit  «^  ^  x^~^  the  limit  a^~\  etc.  Multiplying 
by  a,  (fi,  etc.,  we  see  that  each  term  of  the  second  member 
approaches  the  limit  aP'~'^.  Because  there  are  n  such  terras, 
we  have 


x^ 
Lim.  -^ 


(x=a) 


na^ 


-1 


X  —  a 
Case  II.     When  n  is  a  positive  fraction. 

Suppose  n  ■=^     ,  p  and  q  being  whole  numbers.     Then 

V         p 


x^ 


a' 


_  x9  —  n^ 
X  —  n  X  —  a 

Let  us  put,  for  convenience  in  writing, 


^  =  y, 


then 


and 


X 


f, 


x^  —  a: 


n 


X 


a 


a^  =.!)', 
a  =  b^i 

v^~V  _     >i-b 

y 


b'^ 


LIMITS. 


3(37 


the  limit  ol' 


As  X  approaches  indefinitely  near  to  a,  and  consequently  y 
to  by  the  numerator  of  this  fraction  (Case  I)  aj)[)roaches  to 
pbP~^  as  its  limit  and  the  deuominn,tor  to  qb^r'^.  Hence,  the 
fraction  itself  approaches  to 

qb^-^        q 


Substituting  for  b  its  value  a  ,  we  have 


'to 

x^  —  rt" 

X  —  a 


X'l'  rt»  n 

Lim.  '-^ (^-a)  =  ''-  bP-'J  ='  a  ^ 


p~q 


nd 


n-\ 


Hence  the  same  formulae  holds  when  n  is  a  positive  fraction. 
Case  III.     When  n  is  negative. 

Suppose  n  =.  —p,  ji  itself  (without  the  minus  sign)  being 
supposed  positive.     Then 

xn  —  an       x~P  —  (rP  „  la^  —  x^\ 

— —  x-p  crP  I 1 

X  —  a  X  —  a  \  X  —  a  / 

x^'-a^ 


z=i  —  x~P  a~P 


X  —  a 


I         X  —  a 

I       we  have 

3rs. 

Then 

1 

When  x  approaches  a,  then    x~p    approaciies    a~P,    and 

<^  _  rt^ 

approaches  jya^~^     Substituting  these  limiting  values, 


Lim. 


x^' 


a^ 


(x=o)  =  —  cr^paP-^ 


■2)a-^P~K 


X  —  a 
Substituting  for  —  j9  its  value  n,  we  have 

X^ Qn 

Lim. (x=a)  =  na^~K 

X  —  a 

Hence, 

Theorem.    The  formula 

X^ nTl 

Lim. {x=a)  =  na''^~^ 

X  —  a 

is  true  for  all  values  of  n,  whether  entire  or  fractional, 
positive  or  negative. 


• 


!       !    I        Pi 
i       ■    I 


'.  ni 


n 


.!  'I 


'Hir 


308 


BINOMIAL    TIIEOUKM 


m  . 


ii     I 


CHAPTER     V. 

THE    BINOMIAL    AND    EXPONENTIAL    THEOREMS. 


The  Biiioiuiiil  Tlu'oreiu  for  all  Values  of  i\w 

Exi)oneiit. 

307.  We  have  sliown  in  §§  171,  J3G4,  how  to  develop 
(l+.r)"'  when  n  is  a  positive  whole  number.  We  have  now  to 
ilnd  the  development  when  n  is  negative  or  fractional.    Assunu' 

(1  +  x)»'  =  Bq  -^  BiX  +  B„x^  +  B.^x^  +  etc.,         {a) 

Bq,  B^,  etc.,  being  indeterminate  coellicients.  Because  tliis 
e([uation  is  by  hypothesis  true  for  all  vahies  of  x,  it  will  remain 
true  when  we  put  another  quantity  a  in  place  of  x.     Hence, 

(1  +  aY  —  ^0  +  ^h^''  +  ^2«^  +  ^3^*^^  +  e<^c.  {b) 

Subtracting  {h)  from  (r/),  and  putting  for  convenience 
X  =  1  +  x,        A  —  1  -{■  a, 
the  difference  of  the  two  equations  («)  and  (h)  will  be 
X"  -  A""  =  B^(x  —  a)  +  B.,  {x'^  -  a^)  ^  B^{:i^-  a^)  +  etc. 

The  values  we  have  assumed  for  X  and  A  give 

X—  A  =  X  —  a. 

Dividing  the  left-hand  member  by  X  —  A,  and  the  right- 
hand  member  by  the  equal  quantity  x  —  a,  we  have 


a;2  —  «2 


yn  in 

X—  A  ^  ^  X  —  a  ^  X 


a.^  —  a^ 


a 


+  etc. 


Now  suppose  X  to  approach  a.  The  limit  of  the  left-hand 
member  will  be  7iA^~'^.  Taking  the  sum  of  the  corresponding 
limits  of  the  right-hand  member,  Ave  shall  have 

7iA^-^  =  B^  +  2B^a  +  ^B^a^  +  ^B^a^  +  etc. 

Replace  A  by  its  value,  1  +  «,  and  multiply  by  1  +  a. 
We  then  have 


lilSOMlAL    TllEOnHM. 


m,) 


«(1  -f  nY  =  Bt  (1  +  «)  4-  2/i8rt(l  +  a)  +  ',^,(^^{1  +  ^0 

+  in^a^  (1  -f-  r/)  +  etc. 

Multiplying  the  c'(|iuition  (b)  by  n,  wo  imve 

«  (1  H-  f^'*  =  '^^0  +  ^^B\a  +  ^iA'g^/'^  +  uB^al 

Eqiiatiii^jj  the  cocllicients  of  the  like  powers  of  a  in  these 
iMjiuitious  (§  ;^81),  we  luive,  first, 

By  putting  a  =.  0  in  etiuution  (/>),  we  find  //^  =  1,  whence 

«..=  «^  =  c;). 

Then  we  find  suecessivelv, 
^)i?2  =  {n-\)  B„  whence  Z^g  =  ^^~~^  B^  =  ^*  ^'^  ~  ^^ 


1.^ 


3/;3  =  (71-2)  B,, 


a 


n  —  2        _n{H  —  l){u-2) 


B,  =  V-  B, 


3     ~"^  l.;i.3 

Substituting  these  values  of  Bq,  B^,  />„,  etc.,  in  the  eciua- 
tion  (a)  and  using  the  abbreviated  notation,  v>.,  obtain  the 
e(iuation 

(1  +  xY  =  1  +  nx  4-  (!J)  x'^  +  (!;)  .^3  4-  etc.,  (c) 

which  equation  is  true  for  all  values  of  u. 

308.  There  is  an  important  relation  between  the  form  of 
this  development  when  u  is  a  positive  integer,  as  in  §§  Vti  and 
^01,  and  when  it  is  negative  or  fractional.  In  the  former 
case,  when  we  form  the  successive  factors  7i  —  1,  n  —  2, 
//  —  3,  etc.,  the  ti^''-  factor  will  vanish,  and  therefore  all  the 
( oellicients  after  that  of  .^•"■  will  vanish. 

But  if  n  is  negative  or  fractional,  none  of  the  factors 
■u  —  1,  71  —  2,  etc.,  can  become  zero,  and,  in  consequence,  the 
Scries  will  go  on  to  infinity.  It  therefore  becomes  necessar}^ 
ill  this  case,  to  investigate  the  convergence  of  the  development. 

If  X  >  1,  the  successive  powers  of  x  will  go  on  increasing 

indefinitely,  while  the  coefficients  (   ),   (    ),  etc.,  will  not  go 
24 


t*  »■ 


I    I 


<     •'  IP. 


't    'I'  : 
!    Ay 


370 


niNOMIA  L    TUKOllKM, 


*  !■ 


I. 


<  \Vt 


oil  diniiuishin;.';  indefmitGly  in  tlio  samo  ratio.  For,  let  \\a 
consi  lor  two  Huccessivo  terms  of  the  duvelopiiient,  the  (?'+!)", 
and  tl.o  {i  +  2)"",  namely, 

(;.')..    ana    [.ly^. 

The  quotient  of  the  second  l)y  tiie  first  is 


I    n     \  in\        n  —  / 


As  i  increases  indefinitely,  this  eocfllcient  of  x  will  approach 
the  limit  —  1  (§  '304),  while  x  is  by  hypothesis  as  ^eat  as  1. 
Therefore,  by  continnins^  the  series,  a  point  will  l)e  reached 
from  which  the  terms  will  no  longer  diminish.     Therefore, 

Tlie  development  of  (1  +  .r)"  iii  pojrevs  of  x  is  not  coti- 
vergent  luiless  x  <  1. 

In  consequence,  if  we  develop  {a  +  />)«  when  n  is  negative 
or  fractional,  we  must  do  so  irj  ascending  powers  of  the  lesser 
of  the  two  quantities,  a  or  h. 

EXAMPLES. 

I.  Develop  (1  +  x)^-,  or  the  square  root  of  1  +  a;. 
Putting  n  =  ,- ,  we  have 


(f)  =  ^• 


1(1-1 


) 


1-2 


1:1 

2-4' 


^G-OQ-)   .u 


1.2.3 


2-4-6 


/n\  _  2 (n\  _  _  1-1-3-5 

W  ~      4     \3/  "~       2-4-6-8* 


etc. 


etc. 


etc. 


JilNOMlA  L    TIJHOHKM. 


\Mi 


X  is  not  coil- 


Whence, 
(14.  .;)i  ^  1  +  ^..  _  ^^^^2  +  ^^^^^  -  ----^^^^.^  -I-  ot(.. 

li*  .f  is  II  siiiall   IViiclion,  \hv  terms  in  ^:2,  r\  etc.,  will  bo 

!nueli   .siiuiiler  than  -x  itself,  und   the  first  two  terms  of  tlic 

scries  will  give  ii  result  very  near  the  truth.      We  therefore 
eonoludo: 

T/ir  sqnnrr  root  of  1  f)las  a.  snudl  frncMoih  is  (ij)j)ro.i'i- 
uKitrhj  equal  to  1  plus  luilf  that  fraction. 

2.   To  develop  VlO. 

W^e  sec  at  once  that  VlO  is  between  3  and  4.     We  put  10 
in  tlie  form 

32  +  1  =  3-^(l  +  ^), 

when  VlO  ==  3(1  +  ]^'  • 

Then,  by  the  development  just  performed, 

/         1V^_  ,        J 1         _} 5__ 

U  "^  U/    ~  2.9       8.92  "•"  1G".93       128.94  ^ 

We  now  sum  the  terms  : 


}  t 


1st  term, 

1.0000000 

2d      " 

=  1st    :   18,  .    .    .    . 

+   .0555556 

3d      " 

=  2d     :        3G,  .    . 

.     —  .0015432 

4th     " 

-  3d     :         18,   .     . 

.     +   .0000857 

5th     " 

^  4tli   X        5   :   72, 

.     —  .OOOOOGO 

6th     " 

=  5th   X   —  7  -^  90,     . 

+   .0000005 

Whence, 


4 


Sum  =:  (1  +  ^y  -  1.0540926 
VlO  =  3  X  sum  =::  3.1622778 


which  may  be  in  error  by  a  few  units  in  the  last  place,  owing 
to  the  omission  of  the  decimals  past  the  seventh. 


» ' 


il 


373 


EXPONENT! .  I  A    THEOREM. 


\     f 


3.  To  (levcloi)  V'"^. 

Wo  sue  thiit  3  is  till'  nearest  wliule  number  of  the  root.     S»» 
we  put  

Vs  =  V(:!»-  1)  =  a/:!-'(i  -  ')  =  ;i  (i  -  I)', 

rnjui  which  the  development  may  be  eilected  as  before. 

EXERCISr,  s. 

1.  Compute  the  square  root  of  8  to  0  decimals,  and  from  it 
lind  the  S([uare  root  of  ^  by  §  1H;5. 

2.  Develop  (1  —  .?•)-. 

3.  Develop  (I  —x)'K  iind  I'Xpress  the  term  in  xK 

,       1  1-3   .,  ,    1-3.5   „  ,     . 

Am.    ^+^  +  ^71^+^:^,^^''-^^^' 

4.  Develop r  and  cxi)ress  the  general  term. 

(1  +  ^)^ 

/         1\"* 

5.  Develop  ( 1  +    .)    ii"(l  express  the  general  term. 

6.  Develop  (1  —  xy\  and  express  the  general  term. 

7.  Develop  the  m^^  root  of  1  +  m. 

8.  Develop  {a  —  h)-\  when  a  <,  b. 

9.  Develop  (1  —  x)'»\  when  x  >  1. 

Because  the  development  will  not  be  convergent  in  ascend- 
ing powers  of  x  when  x  >  1,  we  transform  thus: 

1  —X  =  —xll—  -J, 
and  so  put        (1  -  x)-'''  =  (-  x)'"'  \l  -  ^j     • 

10.  Develop  the  in^'''  power  of  1  H 

11.  Compute  the  cube  root  of  1010  to  six  decimals. 


I    *    1 


i:xpfh\7:x TL 1 A   riii:(Hii:sr. 


373 


12.  Dcnclop  {\/n  +  ^/<'>')^ 

13,  Uising  the  riujc'tiunul  notation, 

multiply  tlu'  two  seriL'S,  (p^tn)  uiid  0(y/),  and  sliow  by  tlic  for- 
iiiiiKV  ot'^  •.*<)!  that  the  pruiliict  i.s  e(pial  to  f/>(m  4-  }i). 

The  Exponential  Theorem. 

J501).  Let  it  l)e  ro([uire(l,  if  jxissihle,  to  (U'veh)p  a^''  in 
powers  of  .1',  a  being  any  quantity  whatever.     Assume 

«•'•  =  (\  +  C\x  +  rV^:-  +  C'gi:^'  +  etc.  (1) 

1.)  he  true  tor  all  values  ot'.i^.     Putting  any  other  (luantity  ij  in 
jtlaee  of  .f,  we  shall  have 

a'J  =  a,  +  C\y  4-  (W  +  <"3Z/"  +  etc.  (2) 

By  the  law  of  exponents  we  must  always  have 

Now  the  value  of  a^^y  is  found  by  writing  x  -\-  y  for  .t  in 
(1),  which  gives 

a^^y  =  C\  +  C\  {x+ij)^  C\  {x^iifJr  C^  [x  +  yf^QiQ.     (3) 

On  the  other  hand,  by  multii)lying  equations  (1)  and  {'i), 
we  Ihid 

a^ay  ==  (7,2  +  C,C,y  +  C\C,y^  +  CoC'3/    +  etc. 

+  C\i\x  +     r.'i^A-?/  +  C\C^xff  +  etc. 

+  6;  (73,^2  ^  C'^  (72.i2^  +  etc. 

4-  Co  Cg^r^    _|_  etc. 

By  §  285,  the  coefficients  of  all  the  ])roducts  of  like  powers 
ef./'and  ?/ must  bo  e([ual.  By  equating  them,  we  shall  have 
more  equations  than  there  arc  quantities  to  be  determined, 
and,  unless  these  equations  are  all  consistent,  the  development 
is  impossible.  To  facilitate  the  process  of  comparison,  we 
luive  in  equation  (4)  arranged  all  terms  which  are  homogeneous 
in  X  and  y  under  each  other. 


(^) 


i   »' 


m ' 


',m 


BINOMIAL    THEOREM. 


By  iiutting  x  ■=()  in  (1),  we  find 

u'O  =  6V,         whence         C\  =  1.     (§  103.) 

Comj)a]"ing  the  terms  of  the  first  degree  in  ./•  iind  y  in  (3) 
and  (4),  we  find 

Coefficient  of  a;,  C^  —-  CqC^; 


(I 


II 


y^ 


c. 


C/qCj. 


These  two  equations  are  the  same,  and  agree  with  Cq  —  1 ; 
but  neither  of  them  give^i  a  vaUie  for  6'j,  which  must  tlierefore 
remain  undetermined. 

Comparing  the  terms  of  the  second  degree,  we  find,  by  de- 
veloping {x  +  y)S 

C^  {x^  +  2xy  +  y^)  =  C,x^  +  C.^xy  +  C„jf, 
which  gives  2C„  —  C\% 


whence 


^'  -  1.1  ^''' 


Comparing  the  terms  of  the  third  order  in  the  same  way, 
we  have 

C,  (^  +  3xh/  +  3xy'^  +  y^)  =  (\x^  -f  C^  C\xhj  +  C^  C\xf  +  C,y\ 


whicii  gives 
whence 


Q n    _.-    n  n     —  ^  n  i. 


P    —  - 

'^  -  1.2.3 


_  rr  3 
o  ^1  • 


If  the  successive  values  of  C  follow  the  same  law,  we  shall 
have 


and  in  general. 


n    _  r'  4. 

f  4  —  41^1  ' 


(5) 


Let  us  now  investigate  whether  these  values  of  C  render 
the  equations  (3)  and  (4)  identically  equal. 

Let  us  consider  the  corresponding  terms  of  the  n^^  degree, 
71  being  any  positive  integer.     In  (3)  this  term  will  be 

Cn  {x  -\-  yY- 


ust  tliereforc! 


ne  same  wav, 


law,  we  shall 


EXPONENTIAL    THEOREM. 
Expanding,  it  will  be 


875 


(<;) 


.     In  (4)  the  sum  of  the  corresponding  terms  will  be,  putting 

The  first  terms  in  Hie  two  expressions  are  identical. 
The  comparison  of  the  second  terms  gives 

nOn  =  C\Cn^i,        whence        Cn  =  --  G-i 

This  corresponds  with  (5),  because  (5)  gives 


t'/i-i  — 


(n-l) 


1  rrn~l 


and  if  we   substitute  this  value  of  C..,  in  the  preeedino- ex- 
pression for  Cn,  it  will  become  ^' 

6f  «7f 


^n  — 


71  \ 


which  agrees  with  (5). 

The  third  terms  of  (6)  and  (7)  being  equated  give 

Substituting  the  values  of  C^,  C,,  and  Cn-^  assumed  in  the 
general  form  (5),  v/e  have 

l^^\  J_  r'^  -  1        1  n 

and  we  wish  to  know  if  this  equation  is  true. 

Multiplying  both  sides  by  nl  and  dropping  the  common 
factor  d ,  it  becomes 

(n\  __  n  \ 

\%1        V.  (71  -2)1' 
^vhich  is  an  identical  equation. 

In  the  same  way,  the  comparison  of  the  following  terms  in 
(0)  and  (7)  give 

('^]  ^  ^} ' /n\  _  n ! 

\3/        3 !  (^  _  3) ! '         \47  ~  Tr{^^'4^\ '     ^^^•' 


vi    i 


\    f 


T'l 


376 


EXPONENTIAL    TRiiOREM. 


all  of  which  are  identical  equations.  Hence  the  conditions  of 
the  development,  namely,  tliat  (G)  and  (I),  and  therefore  (;3) 
and  (4),  shall  be  identically  oqnal,  are  all  satisfied  by  the  values 
of  the  coefficients  C  in  (5).  Substituting  those  values  in  (1), 
the  development  becomes 

d'^  =  \^CyX^-^--  C^j?  -f    \  ,  C\^c^  +  etc.         (S) 

X.'  lii  1  '  <i'  O 

This  development  is  called  the  Exponential  Theorem, 
as  the  development  of  {n  +  b)"'  is  called  the  binomial  theorem. 

310.  The  valne  of  C\   u  hUll  to  be  rietermined.     To  do 

this,  assign  to  x  the  particular  valine  -:-,--  •  Then  the  equation 
(8)  becomes  ' 

1  111 

«^>   =   1  +  1  +  :r-7,   +  i-a-.7  +  r-,-  .T-7  +  etc.,    Cfd  iuf,       (0) 

The  second  meml)er  of  this  equation  is  a  pure  number, 
without  any  algebraic  symbol.  We  can  readily  compute  its 
approximate  valne,  since  dividing  the  third  term  by  3  gives 
the  fourth  term,  dividing  thii  by  1  gives  the  fifth,  etc.     Then 


1  +  1  =- 
1-^1.;^: 

1  -=-  1 


o 


1  ~  i.2-;3.4  - 

1  _^  1.2- 3.4. 5  — 

1  _^  1  c.;]-4.5.n  = 

1    ^    1..2.3.1-5-G.T    :rr 

1  ^  ;.2.3.4.o.G.T-8  =: 
1  _!_  1.2.3.4.o.6.r-8.9  — 


2.000000 
.500000 
.1GGGG7 
.041GG7 
.008333 
.001389 
.000198 
.000025 
. 000003 


Sum  of  the  series  to  G  decimals,     2.718282 

This  constant  number  :s  extensively  used  in  the  higher 
mathematics  and  is  called  tne  Naperian  hase."^  It  is  rei)re- 
sented  for  shortness  by  the  symbol  e,  so  that  e  =  2.718282.... 

The  last  equation  is  therefore  written  in  the  form 


e. 


*  After  Baron  Napier,  the  inventor  of  logarithms. 


EXPONENTIAL    THEOREM. 


377 


Raising  to  the  (\^f^  power,  we  luive  a  =  e<^'.     Hence  : 

The  quantity  C\  is  the  e.vpoiipiit  of  tJie  power  to  ivliicli 
we  DUist  revise  the  constant  c  to  produce  the  nmnber  n. 

We  may  assign  one  value  to  a,  nann\v,  e  itg^K,  whicl)  -will 
load  to  an  interesting  result.  Putting  a  =:  c,  we  have  L\  ~\. 
and  the  exponential  series  gives 


X 


T' 


If  we  put  «:=:],  we  have  the  series  for  c  itself,  and  if  wo 
put  X  -—  —  1,  we  luive 


-  =  I 

e 


1 


^  I'^z     1.2.; 


3  "^  1.^.3.1 


—  etc. 


AVe  thus  have  the  curious  result  that  this  series  and  (i»)  arc 
the  reciprocals  of  each  other. 


'■•    i[ 


EXERCISES. 

1.  Substitute  in  the  first  four  or  five  terms  of  the  expres- 
sions (G)  and  (7)  the  values  of  C^^  C\,  Cn-2,  etc.,  given  by  (5), 
aud  show  that  (6)  and  (7)  are  thus  rendered  identical ;y  equal. 

Note.  This  is  merely  a  slijjht  modification  of  the  ])rocess  we  have 
actually  followed  in  comparing  the  coefficients  of  like  powers  of  x  and  y 
in  (0)  and  (7). 

2.  Compute  arithmetically  the  values  of  2.71832,  2.7183-1, 
and  3.7183-2j  and  show  that  they  are  the  same  numbers,  to 
three  places  of  decimals,  that  we  obtain  by  putting  x  =  2, 
X  —  —  1,  and  X  =  —'Z  in  (10),  and  computing  the  first  eight 
or  ten  terms  of  the  series. 

3.  Since  e^'^"'  =  e^,  the  equation  (10)  gives,  by  substituting 
the  developments  of  e^"*"^  and  e''^, 

(1  +  xY       (1  +  .r)3       (1  +  xY 
1  +  1  +  :^  +  —It--  +  -^T--  +  -^TT—  +  etc. 


2! 


3! 


4! 


I  X^         X^         X^  ,      \ 

=  .(l  +  .-  +  ^;  +  5-,  +  j-,  +  etc.) 


It  is  required  to  prove  the  identity  of  these  developmentSf 
by  showing  that  the  coefficients  of  like  powers  of  x  are  equal. 


(    ■ ' 


1 

t 

1 

i' 

378 


LOGARITHMS. 


CHAPTER    VI. 

LOGARITHMS. 


M^' 


I. 


i511.  To  form  the  logarithm  of  a  number,  a  constant  num- 
ber is  assumed  at  pleasure  and  called  the  base. 

Def,  The  Logarithm  of  a  number  is  the  exponent 
of  the  power  to  which  the  base  must  be  raised  to  pro- 
duce the  number. 

The  logarithm  of  x  ig  written  log  x. 

Let  us  put  a,  the  base  ; 

X,  the  number  ; 

I,  the  logarithm  of  x. 


Then 


a^  =z  X. 


Eem.  For  every  positive  value  we  assign  to  x  there  will  be 
one  and  only  one  value  of  I,  so  long  as  the  base  a  remains  un- 
changed. 

Def.  A  System  of  Logarithms  means  the  loga- 
rithms of  all  positive  numbers  to  a  given  base.  The 
base  is  then  called  the  base  of  the  system. 

Properties  of  Logarithms. 

313.  Consider  the  equations, 

(fi  =1',   \  Mogl    =  0; 

a>  =  a\   >•  whence  by  definition,  <  \oga  =  I', 
«^  =  «2 ;  )  (  log  cfi  =  2. 

Hence, 

I.   TJie  logarithm  of  1  is  zero,  whatever  be  the  base. 
II.   TIte  logarithm  of  the  base  is  1. 

III.  The  h'jarltlnii  of  any  number  between  1  and  the 
base  is  a  positive  fraction. 

IV.  The  logarithms  of  powers  of  the  base  are  integers, 
but  no  other  logarithms  are. 


LOGARITHMS. 


379 


-r.  •  i 


onstant  num- 


Again  we  have 


«-i  = 


a-2    = 


1 

a 
1 

rt2 


cr^  = 


a 


n 


Hence, 


log  --    =  -  1 ; 


a 


whence  by  definition,  !  log  -   =  _  o . 


log  —  =  —  n. 
6  a" 


) .. 


.« 


iS  the  loga- 
L  base.    The 


1 

-0; 

a 

1; 

«2 

-  2. 

&e  ^/^e  &(7,s^. 

3Jo 

1  «7^fZ   //^C 

are  integers, 

V.  Tlie  logarithm  of  a  muiiber  hctween  0  and  1  is 
negative. 

Again,  as  we  increase  w,  the  value  of  ««  increases  without 

limit,  and  that  of  —  approaches  zero  as  its  limit.     Hence, 

VI.  TJie  logarithm  of  0  is  negative  inftnity. 

VII.  Theorem.     Tlie  logarithm  of  a  product  is  equal 
to  the  sum  of  the  logarithms  of  its  factors. 

Proof.     Let ;;  and  q  be  two  factors,  and  suppose 

h  =  log  p,  k  =  log  q. 

Tj^en  a^  —  p,  a^  =  q. 

Multiplying,  a^a^  —  a^+f-  =  pq. 

Whence,  by  definition, 

h^-k  =  log  {pq), 
or  log  p  4-  log  (jr  =  log  {pq). 

The  proof  may  be  extended  to  any  number  of  factors. 

VIII.  Theorem.  The  logarithm  of  a  quotient  is  found 
hy  subtracting  the  logarithm  of  the  divisor  from  that  of 
the  dividend. 

Proof.  Dividing  instead  of  multiplying  the  equations  in 
the  last  theorem,  we  have 


.=  a^-^ 


P 

—  < 

9[ 


(1  ■ 


''t  '■■■  \\\ 


I 


^  H 


11  ' 


J  lf^/"f<^^^— «— -nil  <  I  mum. 

Si'' '      


i     ' 


■6, 


380 


LOQARITUMS. 


or 


Hence,  by  definition,        h  —  k  =  log  '\ 

log;;- log  r/  =  log^'. 


IX.  Theorem.  The  logarithm  of  any  power  of  a  num- 
ber in  equal  to  the  logaritlnn  of  the  number  multipl'i'd 
by  the  exponent  of  the  pou'er. 


Proof.    Let  h  =  log  /;,  and  let  w  be  the  exponent. 
Then  «'*  =  p. 

Euiding  both  sides  to  the  )i^^  power, 


a^fi  =  p^. 


Whence 


7ih  =  logj>", 

or  n  log  2^  =  log ;?". 

X.  Theorem.  TJie  logarithm  of  a  root  of  a  number 
is  equal  to  the  logarithm  of  the  number  divided  by  the 
index  of  the  root. 

Proof.     Let  s  be  the  numlier,  and  let  p  be  its  n^^  root,  so 


that 

Hence, 
Therefore, 

or 


p  =^  \/s        and        s  =  ;;"■. 

log  s  =  log  ;;«•  =  n  log  ^j.     (1X1^ 
log  s 


logp  = 


n 


,  n/  l02f  s 

log  V5  —  - 


n 


—     on 


Note.    We  may  also  apply  Th.  IX,  since  p  =  A     Con- 
sidering -  as  a  power,  the  theorem  gives 


n 


^ogp  =  -  logs. 


EXERCISES, 


Express  the  following  logarithms  in  terms  of  hg  p,  hg  q, 
log  X,  and  logy,  a  being  the  base  of  the  system : 


LOGARITHMS. 


381 


ts  n^^  root,  i^o 


1.  Log  p^q. 

2.  Log  jXj^. 

3.  Log/vV 


Alts.  'Z  log  2)  +  log  q. 
4.   Log  })q^xhf. 


5.  Log-  =  log  07;"^  and  cx})lain  the  identity, 

6.  Log  -—  =  lo 

V<1 


1/7-1 


ga-y^;  ^q 
Ans.    Log  .T  +  log  y  —  log ^j  —  log  q. 


7.  L 


OGf 


Log  Va^: 


8.   I 


^0£f  - 


a;"// 


o  ^//ly3 


10.  Log  V^  V^. 


II. 


jO 


or 


;^ 


12. 


Log  V' 


«. 


13-  I^og 

^5-  Log 


rt:r. 


a"' 


1 4.  Loi 


16.   Loo: 


a 

(0 


n-,fn 


17.  Log  Vf(^  —  x' 


A71S. 


18.  Log 


VT 


a^** 


Log  {a  +  ;g)  +  log  ((^  —  rg) 
19.   Log  (a/^  —  .r^). 


Logaritliinic  Series. 

318.  Rem.  The  logaritlim  of  a  number  cannot  he  devel- 
oped in  powers  of  the  number.     For,  if  possible,  suppose 

log  X  =  Cq  -\-  C\x  +  C^x^  +  etc. 
Supposing  x  =:  0,  we  have 

Co  =  log  0, 
HV'hich  we  have  found   to  be  negative   infinity   (§  312,  VI). 
Hence  the  development  is  impossible. 

But  we  can  develop  log  (1  +  ^)  in  powers  of  y.  For  this 
purpose,  we  develop  (1  +  yY  by  both  the  binomial  and  expo- 
nential theorems,  and  compare  the  coefficients  of  the  first 
power  of  X.    First,  the  binomial  theorem  gives 

/-.    ,     s^       -,  xix  —  1)    ^      X  (x  —  1)  (.r  —  2)    o  ,     , 


1-2 


1.2.3 


il     k- 


I      * 
I 

i  ' 

I 


i'! 


!rt 


382 


LOOARirilMS. 


m 


If  wo  develop  the  coefficients  of  ip,  if,  etc.,  by  performing 
the  niuitiplicationa,  we  liave 

.r^  —  X 


Coef.  of  if 


1.2 


part  in  .T  = 


X 

■ 

2 


In  general,  in  the  coefficient  of  ?/",  or 

x{x—l){x  —  'Z)....  (x  —  n  +  1), 
the  term  containing  the  first  power  of  x  will  be 


Hence, 


±l-2-3 (m—  l).r  _       .r 

l')i''3  .  .  .  .  n  ~       u 


(1 4-^)-^'  =  1  +:?•  (y/  -  I  4-  ![-  -  |-  +  etc.)  +  terms  in  x^  A  etc. 

On  the  other  hand,  the  exponential  development,  §  309,  (8), 
gives,  by  putting  1  4-  ;y  for  a. 

(1  +  yY  =  1  +  C^x  -\-  terms  in  x%  x^,  etc. 

Equating  the  coefficients  of  x  in  these  two  identical  series 
we  have 


_       y 


yS 


r 


+  etc. 


(1) 


The  value  of  C^  is  given  by  the  theorem  of  §  310,  putting 
1  +  //  for  a ;  that  is,  Cj  is  here  defined  by  the  equation 

eC,  =:  1  +  y. 

Hence,  if  we  take  the  number  e  (§  310)  as  the  base  of  a 
system  of  logarithms,  we  shall  have 

C^  =  log  (1  +  y). 

Comparing  with  (1),  we  reach  the  conclusion : 

Theorem.    Assuming  the  Kapevian  hase  e  as  a  base, 
iv6  have 

log  {{+y)=y~  t  +  I'  _  t  +  etc.,  ad  inf.        (•^) 


I  •■  ir 


the  base  of  a 


LOOARITIIMS. 


381] 


AA     Lorrarithms  to  the  hasn  e  aro  caHed  Naperian 
Logarithms,  or  Natural  Logarithms. 

Tl.o  appellation  "  natural  "  is  used,  because  this  is  the  simplest  Hy.ste.n 
of  logantlmis.  '  j     v... 

Kem.     The  series  (2)  is  not  convergent  when  y  >  1,  and 
tlierefore  must  be  transformed  lor  use. 

Putting  ~y  iov  y  in  (2),  we  have 

log(l-y)^_^_|^_|^_ete. 
S'  btracting  this  from  (2),  and  noticing  that 
log  (1  +  Z/)  -  log  {l~y)  =  log  J:±-^  (Th.  VIIT), 

we  have         log  ^^  ^  2,  +  f  +  ^  +  etc.  (.) 

Kow  w  being  any  number  of  which  we  wish  to  investigate 
the  logarithm,  let  us  suppose  y  =  -— This  will  give 

^  ft   ~'Y'    J- 

1  +y  _  n  +  1 
1  —  y  ~      n     ' 


whence 


^^^  r-y  =  ^^^  —I-  ^  log  (^^  +  1)  -  log  ^?. 


Substituting  these  values  in  (3),  we  have 

log  {71  +  1)  -  log  n  =  —1—  -i. ? I   ?_ 

6  2;i-M  ^3(2/1  + 1)3^5(27^  +  1)-^ 

+  etc.  (4) 

This  series  enables  us  to  find  log  {n  +  1)  when  we  know 
log  n.  To  find  log  2,  we  put  n  =  I,  which,  because  log  1 
=  0,  gives 

7    5^7     '     ^''^*  / 


log  2  ~  2  ( 


1111 


+ 


+ 


^3    '   3.33  "^  5-35  "^  7-3^ 
Summing  five  terms  of  this  series,  we  find 
log  2  =  0.G93147 


» • 


'III 


t  ,'1 


"  f  :  :  !f| 


fi 


\  1 


384 


LOOAIUTILMS. 


Pill  ting  w  =  2  in  (4),  wo  have 

loft :!  =  log  3  +  a  (^^  +  3 ',.,  +  J-^  +  ±^  +  etc.), 

wiiich  ^nves  log  3  =  1.09801'^. 

Because  9  =  3'^,      log  \)  -—  'Z  log  3  =  53.197224. 

Pnttin;,'  11  =  0  in  (4),  wo  have 

log  -.0  =  log  9  +  'i  (V  +  ^J^.,-  +  _1_.  +  etc.), 

whence  log  10  =  2.302585. 

In  tliis  way  the  Njiperian  logarifiinisorall  numbers  may  1)0 
computed.  It  is  only  necessary  to  compute  the  logaritimis  (<!' 
the  ju'ime  numbers  from  tiie  series,  bec-ause  those  of  the  coni- 
])Osite  numbers  can  be  formed  by  adding  the  logaritlims  of 
their  prime  factors.     (§  312,  Xil) 

81.4,  Definitive  Form  of  the  Exponential  Sericfi.  We  arc 
now  prepared  to  give  the  exponential  series  (§  309,  8)  its  deli- 
iiite  form.     Since  the  coellieient  (\  is  defined  by  the  equation 

c^'<  =  a, 
tlie  quantity  ('  is  tlie  Naperian  logarithm  of  a.     Hence,  the 
exponential  series  is 

«^  =  1  +  ^-'og-'^  +  (£:-l|£«J'  +  {^JfJ^^  +  etc., 

which  is  a  fundamental  development  in  Algebra. 

By  putting  a  =■  e,  we  have  log  «  =  1,  and  the  series  l)c- 
comes  that  for  e^  already  found. 

By  putting  x=zl,  we  have  an  expression  for  any  number 
in  terms  of  its  natural  logarithm,  namely, 

^-■^^     1      +       2!       +       3!       +       41      ^^^^^* 


Comparison  of  Two  Systems  of  Logarithms. 

315.  Put        e,  the  base  of  one  system  ; 
a,  the  base  of  another; 
n,  a  number  of  which  we  take  the  logarithm 
in  both  systems. 


LOG  A  It  IT  JIMS. 


38,') 


:arithms. 


Putting  /  and  /  for  tlie  lu«,'iiritlini8  in  tliu  two  nystcms,  wo 
have 


and  tin'rcforo 


ai'. 


(1) 


Now  ])ut  k  for  the  logarithm  of  a  to  the  base  e.     Then 

e*  =  rt, 

and  raising  both  members  to  the  l'^'^  power, 

c^v  —  iil\ 

Comparing  with  (1),       /  =  kX ^ 


or 


V  =  1  X 


1 


C^^) 


This  C(iuation  is  entirely  independent  of  n,  and  is  therefore 
tlie  same  for  all  values  of  n.     llenee, 

Theorem.  //  we  inulti/)Iij  the  lo^^drUhDh  of  (unj 
number  to  the  base  a  by  t/ir  logdvitJim  of  a  to  the  b(tse  e, 
ice  shall  have  the  lo^arlthiii  of  the  iiumber  to  tue  base  e. 

310.  Although  there  may  be  any  mimbei  of  systei  is 
of  logarithms,  only  two  are  used  in  practice,  namely  : 

1.  The  natural  or  Naperian  system,  base  =  c  — 
2.718282  .... 

2.  The  common  system,  base  =  10. 

The  natural  system  is  used  for  purely  algebraic 
purposes. 

The  common  system  is  used  to  facilitate  numerical 
calculations. 

Assigning  these  values  to  e  and  a  in  the  preceding  section, 
the  constant  k  is  the  natural  logarithm  of  10,  which  we  have 
found  to  be  2.302585. 

Therefore,  by  (2),  for  any  number, 

uat.  log  =  common  log  x  2.302585, 


and 


nat.  log. 
common  log  =  ^;^^^^^ 


Hence, 


=  nat.  log  X  0.4342944.... 


25 


|i    » 


1  ■ 


I  ■ 


I' 


ii' 


386 


LOaMUTlIMS. 


Titkouj:m.  77//!  ronininn  In^nrifhin  of  ftnij  nunihrr 
vufi/hr  J'onyiid  hij  inidtiitliiiiid  ifs  imturdl  hnjnrithni  hif 
OA'.U'V.m  ....  //;•  ////  ////'  reciprocal  of  the  JWiftcr'nin  loi^i- 
rif/ini,  itf  10. 

Jhf.  The  number  0.4842944  is  called  the  Modulus 
of  the  eoiniiioii  system  of  logjirithms. 

EXERCISES. 

1.  Show  that  if  a  and  b  he  any  two  hascs,  the  lop^arithm  of 
a  to  the  haso  b  and  tho  logarithm  of  h  to  the  base  a  are  the  n- 
ciprocals  of  caeh  otlier. 

2.  Wliat  docs  this  theorem  express  in  the  case  of  the  natu- 
ral and  common  systems  of  logarithms  ? 


I, 


Coiniiioii  Lo^iii'idims. 


317.  Because 

10'^  =  100, 
10^  =  10, 
100    _       1, 


10-1  ^ 


1 
10' 

10-2  —  __L- 

^"      -  100' 
etc. 


we  have  to  base  10, 


log  100  " 

2, 

log    10  = 

I, 

log      1  _ 

0, 

,        1 
log   1,-   - 

-1, 

^'^  rJo  = 

--^ 

etc. 


The  following  conclusions  respecting  common  logarithms 
will  be  evident  from  an  inspection  of  the  above  examples : 

I.  Tlie  logarithm  of  any  number  between  1  and  10  is 
a  fraction  between  0  and  1, 

II.  The  lo^aritlnn  of  a  number  with  two  dibits  is  1 
plus  some  fraction. 

III.  ///.  general,  the  logarithm  of  a  number  of  i  digits 
is  i  —  1,  plus  some  fraction. 

IV.  The  logaritlini  of  a  fraction  less  than  unity  is 
negative. 

V.  Tlie  logarithms  of  two  numbers,  the  reciprocal  of 
ea,cli  other,  are  equal  and  of  opposite  signs. 


UXiMilTllMS. 


a87 


of  the  luitu- 


10  = 

2, 

0  = 

1, 

1  = 

«), 

VI.  ff  one  nmnhrr  is  lo  times  another,  its  toijfn'illini, 
will  he  jjreafej'  Inj  ituih/. 

Proof.     If  10'  =  w, 

th.Mi  10'»'  -  lux  W  —  10/<. 

lIl'IK'O,   if  /  —    lt»<r  n, 

I  hell  I  -^   {    —   V)\r   10//. 

1518.  To  f^Ive  nri  idea  <»r  the  prn^riVHHion  of  lo^^aritlnnM,  tlio 
I'oUowiiif?  liible  of  lo<5}irillmis  ol'  the  linst  II  mitnluM's  sliould  ho 
studiod. 

The  loji^tirithms  are  not  exuct,  hccauso  all  loorurithnis,  ex- 
cept those  of  powers  of  10,  arc  Irratlojiai  iiuinhcrs,  and  liuTc- 
I'ore  when  expressed  as  decituals  extend  oiit  indefinitely.  Wo 
jj^ive  only  tlio  first  two  deeinials. 

locr  1    =   0.00,  lorr  ;      =   0.85, 

log  2  =  0.30,  log  S    =  0.00, 

log  3  =  0.48,  log  0     =  0.05, 

log  4  =  0.(50,  leg  10  =   1.00, 

log  5  =  0.;0,  log  U  =  1.04. 
logo  =  0.78, 

It  will  be  noticed  that  the  difference  between  two  consecu- 
tive logarithms  continually  diminishes  as  the  numbers  increase. 
F(>r  instance,  the  difference  between  log  W  and  log  10  must 
})y  §  'M'Z,  VIII,  be  the  same  as  between  log  1  and  log  2. 

1511).  Computation  of  Logarithms.  Since  the  logarithms 
of  all  composite  numbers  may  be  found  by  adding  the  loga- 
rithms of  their  factors,  it  is  only  necessary  to  show  how  the 
logarithms  of  prime  numbers  are  computed.  We  have  already 
shown  (§  313)  how  the  natural  logarithms  may  be  comi)ute(i, 
and  (§  310)  how  the  common  ones  may  be  derived  from  them 
by  multiplying  by  the  modulus  0.43421)44....  It  is  not  how- 
ever necessary  to  multiply  the  whole  logarithm  by  this  factor, 
but  we  may  proceed  thus: 

We  have,  putting  M  for  the  modulus, 

com.  log  n  =  M  nat.  log  n, 

com.  log  {n  -\-  \)  —  M  nat.  log  (n  +  1)  ; 


''  II 


I! 


388 


LOGARITHMS. 


whence,  by  subtraction, 

com.  log  (w  +  1)  —  com. log??,  z=z  il/'[nat,  log(w  +  l)  —  nat.log?^]; 

and,   by  substituting    for     nat.  log  (m  4-  1)  —  nat.  log  n     its 
value,  §  313, 

com.  log  {n  +  1)  =  com.  log  u  -f  'ZM 


^1 I  _^ 


By  means  of  this  series,  the  comjiutations  of  the  successive 
logarithms  may  be  carried  to  any  extent. 

Tables  of  the  logaritluns  of  numbers  up  100,000,  to  seven  places  of 
decimals,  are  in  common  iise  for  astronomical  and  matliematical  calcubi- 
tions.  One  table  to  ten  decimals  was  published  about  the  beginning  of 
the  ])resent  century.  The  most  extended  tables  ever  undertaken  wore 
constructed  under  the  auspices  of  the  French  government  about  1795,  and 
have  been  known  under  the  name  of  Les  Grander  Tables  du  CaddHtn. 
Many  of  the  logarithms  were  carried  to  nineteen  places  of  dccimalH. 
They  were  never  published,  but  are  preserved  in  manuscript. 

Ji20.  It  may  interest  the  student  who  is  fond  of  computa- 
tion to  show  how  the  common  logarithms  of  small  numbers 
may  bu  comiiuted  by  a  method  based  immediately  on  first 
principles. 

Let  n  be  a  number,  and  let  -  be  an  approximate  value  of 

its  logarithm.     We  shall  then  have, 

n  =  10^, 
or,  raising  to  the  q^^  power, 

n'i  =  lOP. 

Hence,  could  we  find  a  power  of  the  number  which  is  also 
a  power  of  10,  the  ratio  of  the  exponents  would  at  once  give 
tlie  logarithm.  This  can  never  be  exactly  done  with  whole 
numbers,  but,  if  the  condition  be  approximately  fulfilled,  we 
shall  have  an  approximate  value  of  the  logarithm. 

Let  us  seek  log  2  in  this  way.  Forming  the  successive 
powers  of  2,  we  find 

2^0  =  1024  =  103(1.024).  (1) 


LOGARITHMS. 


389 


the  successivo 


mate  value  of 


the  successive 


Hence,    3  :  10  —  0.3    is  an  approximation  to   log  2.     To 

find   a  second  approximation,   we  form  the  powers  ot!  1.0'M 

until  we  reach  a  numl)er  nearly  equal  to  2  or  10,  or  the  (|Uo- 

tient  of  any  i)ower  of  2  hy  a  power  of  10.     Suppose,  for  instance, 

that  we  find 

1.02-1'^'  =  2. 

Because  1.024  =  2^^  4-  10^,  this  equation  will  give 


2,     or     2^0'" -- 2.103^,     or     2i"-'^-i  =  10">^, 


which  will  give 


log  2  = 

^  102'  - 


If  we  form  tlie  powers  of  1.024  by  the  binomial  theorem, 
or  in  any  other  way,  we  shall  find  that  x  is  between  29  and  30, 
from  which  we  conclude  that  log  2  =  0.301  nearly. 

To  obtain  a  yet  more  ex-ict  value,  we  form  the  30th  power 
of  1.024  to  six  or  seven  decimals,  and  put  it  in  the  form 

1.024-*  —  2  (1  +  a), 

where  «  will  be  a  small  fraction. 

Then  Ave  findwdiat  power  of  1  +  «  will  make  2.     L(>t  y  be 

this  power.     Raising  the  last  e<|uation  to  the  ijth  power,  we 

luive 

1.02430y  -  2i'(l  +  a)y  =  2i'*i. 

Putting  for  1.024  its  value,  2^*^  -^  10^  this  equation  l)ecomes 

2300y 


10902/ 


—  07/ n 


or 


whence. 


log  2  =  o 


lO^J, 


299//  —  1 

By  a  little  care,  the  value  of  y  can  be  obtained  so  accurately 
that  the  vahie  of  log  2  shall  be  correct  to  8,  9,  or  10  }tlaces  of 
decimals. 

The  power  to  which  we  must  raise  1  +  r «  to  produce  2  will 

,  .      .  1    Nap-  log  2       ,  .  „ 

be  approximately  — - — - — ,  when  «  is  very  small. 


■I' 


I     t 


i      i 


■'■  r 


890 


LOGAPJTRMS. 


\ ,  II 


EXERCISES. 

1.  In  the  common  system  {a  —.  10)  we  have 

log  2  =  0.30103,  log  8  ~  0.47712. 

Ilcncc  find  the  logarithms  of  4,  5,  0,  8,  9,  12,  12|-,  15,  in, 
inf.  18,  20,  250,  0250. 

Note  that  5  =  V,  '^^  ~  H^>  1Q§  =  -'},",  and  npply  Tu.  VIII. 

2.  How  many  digits  are  there  in  the  Inindredth  power  of  "^  ? 

3.  Given  log  49  =  1.690190  ;  find  log  7. 

4.  Given  log  1331  —  3.124178;  find  log  11. 

5.  Find  tlie  logarithm  of  105  and  1,05  from  the  above  data  ? 

6.  Find  the  logarithm  of  1.05'*^. 

7.  If  $1  is  put  out  at  5  per  cent,  per  anLum  compoiiiid 
interest  for  1000  years,  how  many  digits  will  be  required  to 
express  the  amount?     (Compare  §  210.) 

8.  Prove  the  equation 

1  1 

log  .7;  =r  -  log  {X  i-  1)   +  ^  log  {X  -  1) 

r__j^ i_  _         1 

"^    L''^^'*^'"^^  "^  H^h^-'iy  '^'  5  {2xi  -  ly  "^  ^^^• 

9.  If  If  —.  log  .?,  of  what  iiumbers  will  y  -{-  1,  7/  -^  2,  y  —  1, 
and  y  —  2  be  the  .'ogarithms  ? 

10.  Find  X  from  the  equation  c^'  —  //. 

Solution.     Taking  the  logarithms  of  both  members,  we  have 

X  log  c  =  log  h  ; 
log  A 
logc 


whence, 


X 


II. 


"OJ' 


71. 


12. 


(PX  ^    1. 
Ill 


13.       h^ 


1 
P 


14.     h~''  =  p. 

Sliow  that  the  answers  to  (13)  and  (14)  are  and  ought  to  be  identical. 
15.      (f<-'^^  =  m.  16.      b(F'  =  i\ 

17.     Find  X  and  y  from  the  equations 

a^'by  —  p,  lOJl"^-  —  q. 


BOOK    XII. 
IMA  G  IN  A  RY     O  U  A  N  TITIR  S. 


e  above  data  ? 


to  1)6  identical. 


CHAF'TER    I. 

OPERATIONS    WITH    THE    IMAGINARY    UNIT.* 

3^1.  Since  the  square  of  either  a  negative  or  a  positive 
quantity  is  always  positive,  it  follows  that  if  we  have  to  extract 
the  square  root  of  a  negative  quantity,  no  answer  is  possible, 
ill  ordinary  positive  or  negative  numbers  (§§  170,  200). 

In  order  to  deal  with  such  cases,  mathematicians  liave  been 
k'd  to  siqrpose  or  imagine  a  kind  of  numbers  of  which  the 
sijuares  shall  be  negative.  These  numbers  are  called  Imagi- 
nary Quantities,  and  their  units  are  called  Imaginary 
Units,  to  distinguish  them  from  the  crdinary  positive  and 
negative  quantities,  which  iire  called  real. 

333.  T/ie  Imaginary  Unit.  Let  us  have  to  extract  the 
square  root  of  —  9.  It  cannou  be  equal  to  -f  3  nor  to  —  3, 
because  the  square  of  each  of  these  quantities  is  +  0.  We 
may  therefore  call  the  root  V —  9,  just  as  we  put  the  sign  -v/ 
liL'tbre  any  other  quantity  of  which  the  root  cannot  be  extracted. 
But  the  root  may  be  transformed  in  this  way  : 

Since  —  9  =  -f  9  x  —1, 

it  follows  from  §  183  that 

a/— ~9  —  a/9  V— "l  =  3\/~l. 


*  It  ia  not  to  be  expected  that  a  beginner  will  fully  understand  this 
subject  at  once.  But  he  should  be  drilled  in  the  mechanical  process  of 
operating  with  imaginarios,  even  though  he  does  not  at  first  understand 
their  significance,  until  the  subject  becomes  clear  through  fauiiliarity. 


>  • 


I  ! 


no2 


TMAGTNA RT   Q UA NTTTIK8. 


it 


Def.  The  surd  V— 1  ia  the  Imaginary  Unit.  Tlie 
imaginary  unit  is  commonly  expressed  by  the  symbol  /. 

This  symbol  is  used  because  it  is  easier  to  write  i  than 
V-l. 

The  unit  i  is  a  supposed  quantity  such  that,  when  squared, 
the  result  is  —  1. 

That  is,  i  is  defined  by  the  equation 

i2  =  -  1. 

Theorem.  Tlie  square  root  of  any  negative  quantity 
maij  he  expressed  as  a  nnniher  of  inuiginary  units. 

For  let  —  n  be  the  number  of  which  the  root  is  required. 

Then  V—  n  =  V  -\-  n  V—  1  =  \^ ni. 

Hence, 

To  extract  the  squa.re  root  of  a  negative  quantihj, 
extract  the  root  as  if  the  quantity  were  positive,  and 
affix  the  syjuhol  i  to  it. 

323.  Complex  Quantities.  In  ordinary  algebra,  any  num- 
ber may  be  supposed  to  mean  so  many  units.  7  or  a,  for 
example,  is  made  up  of  7  units  or  a  units,  and  might  be  writ- 
ten 7-1  or  «1. 

When  we  introduce  imaginary  quantities,  we  consider  them 
as  made  up  of  a  certain  number  of  imaginary  units,  each  repre- 
sented by  the  sign  «',  j'lst  as  the  real  unit  is  represented  by  the 
sign  1.    A  number  h  of  imr.ginary  units  is  therefore  written  bi. 

A  sum  of  a  real  units  and  b  imaginary  units  is  written 

a  +  bi, 
and  is  called  a  complex  quantity.     Hence, 

Def.  A  Complex  Quantity  consists  of  the  snm  of 
a  certain  number  of  real  units  plus  a  cei'tain  number  of 
imaginary  units. 

D(f.  When  any  expression  containing  the  symbol 
of  the  imaginary  unit  is  reduced  to  the  form  of  a  com- 
plex quantity,  it  is  said  to  be  expressed  in  its  Normal 
Form. 


IMA  GIN  A  li  Y   Q  JJA  NTITIbJS. 


;J03 


Addition  of  Comijlex  Expressions. 

334:,  The  algebraic  operations  of  addition  and  subtraction 
are  performed  on  imaginary  (juantities  according  to  nearly  the 
same  rules  which  govern  the  case  of  surds  (§  181  j,  the  surd 
being  replaced  by  i.     Thus, 

aV—  1  +  bV—  1  =  <(i  +  bi  =  (a  +  b)  i. 

Hence  the  following  rule  for  the  addition  and  sul)traction 
of  imaginary  quantities : 

Add  or  subtract  all  the  real  terms,  as  in  ordinal-!/ 
algebra.  Then  add  the  coefficiejits  of  the  iina^i/iftrij 
unit,  and  afji.v  the  symhol  i  to  their  sum. 

Example.  Add  a  +  bi,  G  +  Ti.  5  —  lOi,  and  subtract 
0(1  —  :ibi  +  z  from  the  sum. 

Wo  may  arrange  the  work  as  follows : 

a  +     bi 
0  +     7i 
5  —  iOi 
—  z  —  oa  -f-  2bi     (sign  changed). 

Sum,      —  z  —  U  +  11  +  (S^*  —  3)  i. 

EXERCISES. 

1.  Add  3;/'  H-  42/i  +  m,  "2m  -\-  bni,  Gni  —  Gyi. 

2.  Add  4ai,  17?*,  oa  +  Gbi,  x  +  yi. 

3.  From  the  sum  a  -\-  bi  -\-  m  —  ni  — })  -\-  qi  subtract  tho 

.sum   -\-  yi  —  z  —  ui. 

Txcduce  to  the  normal  form : 

4.  a  -{-  bi  —  (m  —  7ii)  —  {x  ■}-  yi). 

5.  m  {a  —  bi)  —  71  {x  —  yi). 

Multiplication  of  Complex  Quantities. 

335.  Theorem.  Ml  the  even  powers  of  the  ima^i- 
narij  unit  are  real  units,  and  all  its  odd  powers  are 
iniaginarrj  units,  positive  or  negative. 


•-,    ( 


.  ifl 


iil 


5394 


IMA  GIN  A  R  Y   Q  UANT1T1E8. 


Proof.  The  imaginary  unit  is  by  definition  such  a  symbol 
as  when  squared  will  make  —  1.     Hence, 

i^  =  -  1. 

Now  multiply  both  sides  of  this  equation  by  i  a  number  of 
times  in  succession,  and  substitute  for  each  power  of  i  its  value 
given  by  the  preceding  equation.     We  then  have 

i^  =  —  ?', 

i'^  =  —  i^  =  +  1  (because  i^  =  —  1), 


,;5  — 


V  =  —  (^  =  4-  /•'  =  —  i, 


etc.      etc. 


etc. 


It  is  evident  that  the  successive  powers  of  i  will  always 
have  one  of  the  four  values,  i,  —  1,   —  i,  or   -\-  1. 


i, 

i', 

i', 

etc.. 

will  be  equal  to 

• 

^^ 

/^ 

z'lo. 

etc., 

a                    a 

-I; 

i^, 

i\ 

i'\ 

etc., 

a                    a 

—  i', 

i\ 

i\ 

i'', 

etc., 

a                   a 

+  1. 

We  may  express  this  result  thus: 
//  n  is  any  integer,  then: 


{4n 


q'in+S  __ 


1,         i^n+l  -    i^         Hn+2  —   —1, 

To  multiply  or  divide  imaginary  quantities,  we  proceed  as 
if  they  were  real  and  substitute  for  each  power  of  i  its  value  as 
a  real  or  imaginary,  positive  or  negative  unit. 

Ex.  I.     Multiply  ai  by  xi. 

By  the  ordinary  method,  we  should  have  the  product, 
axi^.     But  f  —  —  1.     The  product  is  therefore  —  ax. 


That  is, 


ai  X  XI  =z  —  ax. 


Ex.  2.  Multiply  a  +  hi  by  w?  +  7ii. 

711  {a  +  bi)  =  ani  —  hi  (because  ni  x  hi  ■=.  —hn) 
in  {a  +  hi)  =z  hmi  +  am 

{m  +  ni)  {a  +  bi)  =  am  —  hn  -{-  (an  +  bm)  i, 

which  is  the  product  required. 


IMA  GIN  A  RT   Q  UA  N  TITIES. 


395 


i 

will 

alw 

ays 

1. 

*; 

I; 

*■; 

+ 

1. 

EXERCISES 

Multiply 

I.  X  ■}-  yi  by  a  —  Ik  2.  m  4-  7//'  by  ai. 

3.  ;>i  —  /u'  by  bi.  4.  1  +  /  by  1  —  i. 

5.  X  —  yi  by  «  +  bi.  6.  .^■  —  ?//  by  x  +  y/. 

7.  a  —  ai  —  bi  by  a  +  ai  -\-  bi. 
Develop 

8.  (a  4-  5«y.  9.  (wi  4-  ni)^. 
10.  (1  +  iy-  II-  U  —  '7- 

• 

826.  Imaginary  Factors.  The  introduction  of  imaginary 
units  enables  us  to  factor  exjiressions  which  are  prime  wlien 
only  real  factors  are  admitted.  The  following  are  the  princi- 
pal forms : 

«2  j^  y..  =.  [a  -\-  bi)  {a  —  bi\ 

a^-b^±  2abi  =  {a  ±  bi)\ 

The  first  form   snows  that  the  sum  of  two  squares  can 
always  be  expressed  as  a  product  of  two  complex  factors. 
Por  example,  17  :=  4-'  4-  1<'  =(4  4-  0  (^  —  0- 

EXERCISES. 

Factor  the  expressions : 

I.     x^  4-  4.  2.     X-  +  2. 

3.  !t^  —  2x  4-  5  =  (.,--1)2  +  4. 

4.  X-  —  4:X  4-  13.  5.     a  4-  b. 

6.     a^  4-  2an  4-  5u^.  7.     x"^  -\-  2xy  4-  2yi 

331.  Fundamental  Prixciple.    .4  complex  quantity 
A  4-  Bi  cannot  he  equal  to  zero  unless  we  have  both 

A  =  0        and        B  —  0. 

Proof.     If  A  and  B  were  not  zero,  the  equation  A  -j-Bi  ==  0 
would  give 


:  I  !* 


•*  i*     ; 


!i 


'1 


*=-5' 


that  is,  the  imaginary  unit  equal  to  a  real  fraction,  which  is 
impossible. 

Cor.    If  both  members  of  an  equation  containing  imagi- 


\ 


I  •  J 


390 


IMAGINARY  QUANTITIES. 


'u 


nary  units  are  reduced  to  the  normal  form,  so  that  the  equation 

shall  be  in  the  tbnu 

A  +  /;/  =  J/  f  Ni, 

Avc  must  have  the  two  e({uatious, 

A  =  M,  B  z=  NT. 

For,  by  transposition,  we  obtain 

^  -J/+  {B-N)i  =  0, 
whence  the  theorem  gives  A  —  31  =  0,  B  —  N=0.     Hence, 

Every  equation  hetween  coinjjJex  qiuuUiUes  liivoli'cx 
tuH)  equations  between  real  quantities,  fornied  hy  equatin<J 
the  numbers  of  real  and  imaginary  units. 


lieductioii    of  Functions   of  i   to   the  Normal 

Form. 

3!38.    1.  If  we  have  an  entire  function  of  /, 

a  +  hi  -\-  ci^  +  di^  +  ei^  -\-fi^  +  etc., 
we  reduce  it  by  putting 

and  the  expression  will  become 

{a  —  c  -{-  e  —  etc. )  -j-  [b  —  d  +f—  etc. )  / ; 
which,  when  we  put 

X  =  a  —  c  -{■  e  —  etc.,        y  =  h  —  d  -\-f  —  etc., 
becomes  x  +  yi,  as  required. 

2.  To  reduce  a  rational  fraction  of  /  to  the  normal  form, 

we  reduce  both  numerator  and  denominator.     The  fraction 

will  then  take  the  form 

a  +  bi 

m  +  ni 
Since  this  is  to  be  reduced  to  the  form  x  +  yi,  let  us  put 

a  +  bi  ,      . 

— -— •  =  X  ■{■  yi, 
m  +  ni  ^ 

X  and  y  being  indeterminate  coefficients. 

Clearing  of  fractions, 

a  -\-  bi  :=  mx  —  ny  +  {wy  +  iix)  i. 


hVA  GIN  Alt  Y   q  VANTITIE8. 


807 


Comparing  tlie  iiunibur  of  roul  and  imaginary  units  on 
cacli  side  of  the  equation,  we  have  the  two  e((uatious 

mx  —  ny  =  a,  nx  +  my  =  b. 

Solving  them,  we  find 


X  = 


ma  ■\-  nh 


y 


mb 


iia 


rj..       n  rt  4-  bi        ma  +  7ib    ,   mb 

Therefore,     — ; — .  =  — r— — ^  +  -  ., 


71  a 


I, 


m  -\-  ni        m^  +  /i^        ni'^  +  n^    ' 
whieh  is  the  normal  form. 


le  Normal 


EXERCISES. 

Reduce  to  the  normal  form  : 
I.     7  _  3i  _  Oi"2  +  2/3  +  i^  -  ^•5. 


2. 


7- 


1  +  'i- 

C  —  5/:' 
1-i 

2  +  4^' 


1   +   A 


5- 

8. 


1  —  4 

a  4-  i/ 


3. 
6. 


i-  1 

wu'  (.f  —  ai) 

X  +  rti 

(rt  +  ii)  {a— hi) 


a  —  bi 

10.  What  is  the  value  of  the  exponential  series  which  gives 
the  development  of  e*?     We  put  x  =  i  in  §  310,  Eq.  10. 

11.  Develop  (1  +  xi)^  by  the  binomial  theorem. 

12.  What  are  the  developed  values  of 

(1  +  bi)»  +  (1  -  ^'0" 
and  (1  +  bi)^  —  {1  —  bi)^? 

13.  Write  eight  terms  of  the  geometrical  progression  of 
which  the  first  term  is  a  and  the  common  ratio  i. 

14.  Find  the  limit  of  the  sum  of  the  geometrical  progres- 
sion of  which  the  first  term  is  a  and  the  common  ratio  -• 

329.  To  reduce  the  square  root  of  an  imaginary  expres- 
sion to  the  normal  form. 


Let  the  square  root  be  \/a  4-  bi. 

We  put  X  -\-  yi  =  V«  +  U. 

Squaring,      x^  —  y^  -{-  2xyi  =  «  -f  bi. 


!     I 


'■  M 


t ' 


I     ■ 


,  ■'  I 


¥: 


yys 


'I 


t 


V  » 


IM.  1 67iV.  1  uy   q  UANTITWS. 


Comparing-  units,      x^  —  ip  =  n, 

%ry  —  I). 

Solving  tliis  pair  of  quadratic  e([uation.s,  we  Unci 

_  V_{V¥Tt/^  +  a) 


Therefore, 


^/ai  4-  i2  _  a 


V 


EXERCISES. 

Reduce  the  square  roots  of  the  following  expressions  to  tlu' 
normal  form : 

I.     3  +  4i.  2.     4  +  3i.  3.     12  4-  5i. 

4.  Find   the   square   roots  of   the  imaginary  unit  i,  and 

of  —  i,  and  prove  the  results  by  s({uaring  them. 

Note  that  this  comes  under  the  preceding  form  wlien  «  =  0  and 
6=±1. 

5.  Find  the  fourth  roots  of  the  same  quantities  by  extract- 
ing the  square  roots  of  these  roots. 

8.30.  Quadratic  Equations  ivith  hnaginanj  Roots.  The 
combination  of  the  preceding  operations  will  enable  ns  to  solve 
any  quadratic  equation,  whether  it  does  or  does  not  contain 
imaginary  quantities. 

Example  i.    Find  x  from  the  equation 

x'  +  4a:  4-  13  =  0. 

Completing  the  square  and  proceeding  as  usual,  we  find 

a;2  +  4-2:  +  4  =  —  9, 

whence  a;  -f  2  =  V—  9  =  ± 

and 

Ex.  2. 

Completing  the  square, 


3/. 


X  =  —  2  ±  3*. 
x^  +  bxi  —  C  z=:  0. 


,      ^   .      b^  ^ 

x^  -\-  bxi  —  -  =r  r  —  -  • 
4  4 


IMA  GIN  A  JiV   Q  UANTITIES. 


31)U 


Extracting  the  root, 


.r 


^   hi  _  V4c~  b^ 


'Z 


2 


wlionce 


X 


=  ±^V(4.-/;^)-^. 


±lsi!i)  i 


•essions  to  tlio 

12  -f-  5t. 

y  unit  i,  aiitl 

vhen  a  =  0  and 
ics  by  extnu't- 

Roots.  'riio 
jle  us  to  solve 
s  not  contain 


lal,  we  find 


EXERCISES. 

Solve  the  quadratic  e([uatioii8: 

I.     x^  -^  X  -\-  I  =  0.  2.     .^2  _  a;  -(-  1  —  0. 

3.     a-2  +  32-  4-  10  =  0.  4.     a:2  _^  iq^.  ^  34  _  0. 

Form  quadratic  equations  (^  109)  of  wliicli  tlie  roots  shall  he 
5.     a  4-  hi  and  a  —  hi.  6.     ai  -\-  h  and  ai  —  h. 

3.'51.  Expnnetitial  Fii7icfi(ms.  When  in  the  exponential 
function  a^  we  suppose  2  to  represent  an  imaginary  expression 
./•  4-  yi,  it  becomes 

This  expression  could  have  no  meaning  in  any  of  our  pre- 
vious definitions  of  an  exponent,  because  we  have  not  shown 
what  an  imaginary  exponent  could  mean.  But  if  we  suppose; 
the  effect  of  the  exponent  to  be  detined  by  the  exponential 
theorem  (§§  309,  314),  we  car  develop  the  above  exi)rcssion. 
First  we  have,  by  the  fundamental  law  of  exponents, 

f^x+yi  —  a^avK 
Xext,  if  we  put  c  =  Xap.  log  a,  we  have 

a  =  e^; 
whence,  av^  =  e<^*. 

If  we  put,  for  brevity,  cy  =  u,  we  shall  now  have 

The  value  of  a^  being  already  perfectly  understood,  we 
may  leave  it  out  of  consideration  for  the  present,  and  investi- 
gate the  development  of  e"*.  By  the  exponential  theorem 
(§  310,  10), 

,  ^  .  ?/2^*3  ^^3|;3  ^^4|4  ^/5^'5 

«"'  =  !  +  »'  +  Tr  +  "3 !  +  IT  +  Tr  +  "'"• 


» }• 


1    I 


>! 


'i:  it 


I       ♦ 


'If 


400  l.VA(JL\AIiy    qU  AN  TIT  IKS. 

Siibstituliii<;  for  the  powers  of  /  their  values  (§  325), 

•^  =  1  -  ai  +  41  -  0"!  +  ^•'"-  +  i»  -  a !  +  5!  -  "'^-l  '• 

These  two  series  are  eaeh  fuiiclions  of  ii,  to  which  special 
namen  have  been  given,  namely : 


u* 


It; 


n 


6 


ir 


Dof.    The  series    1  —  j-i  +  ji  ~  Ft  +  ut  ~  ^^^''  ^^  called 

^»  "Xt  ')•  0» 

the  cosine  of*/,  and  is  written  cos  n. 


w 


11° 


w 


ti^ 


Def.     The  series  ^^  —  ni  4-  r  ,  —  ^Tf  +  (Ti  ~  ^^^'^   ^^  tjalled 
the  sine  of  vf,  and  is  written  sin  it. 

Using  this  notation,  the  above  development  beeomos, 


f,Ul    — 


COS  11  +  /  sin  n, 


{") 


■whieh  is  a  fundamental  e<puition  of  Algebra,  and  should  1m' 
memorized. 

Remarks.  These  functions,  cos  u  and  sin  ?^  have  an  ex- 
tensive use  in  both  Trigonometry  and  Algebra.  To  familiarixc 
himscir  with  them,  it  will  be  well  for  the  student  to  compuir 
their  values  from  the  above  series  for  w  =  0.25,  i«  =  0.5(i, 
M=],w  =  2,  to  three  or  four  ])laces  of  decimals.  This  c;ui 
be  done  by  a  process  similar  to  that  employed  in  computiug  c 
in  §  lUO.     If  the  work  is  done  correctly,  he  will  find: 


1 


1 


For 

''       4' 

cos  7  = 

4 

0.9G0, 

sin  J  -  0.247. 
4 

a 

1 

c«s^_ 

0.878, 

sin  I  =  0.479. 

it 

n  =  1, 

cos  1  — 

0.540, 

sin  1  -  0.841. 

a 

n  =  2, 

cos  2  = 

0.410, 

sin  2  -  0.900. 

3«>3.  Let  us  now  investigate  the  properties  of  the  function- 
cos  u  and  sin  u,  which  are  detined  by  the  equations, 


^^2  ^fi  iff6 

cos»  =  l-^  +  ;,,--j  +  etc. 

7i^         V^         U' 

sm  u  —  u  —  —  +  -  —  --  -f-  etc. 


{l>) 


IMA  (USA  U  y   q  UAMlTlEfi. 


401 


Sinco  cos  H  IucIikK's  otily  even  powiTs  of  i/,  its  vuliir  will 
remain  uiichiin^ad  when  we  cluiiigi'  flic  sign  of  n  IVoiii  -}-  lo 


— ,  or  vice  i'i'r.s((. 


11 


ence. 


cos  (—  ?/)  =  c 


08  If. 


(I) 


Since  sin  u  contains  only  odii   powers  of  it,  its  si«^n  will 


clmnire  with  that  of  u.     Hence, 


sin 


(- ")  =  - 


sni  II. 


{'^) 


Tf  in  the  cqnation   (a)  we  chnni^c  the  sign  of  ?/,  wc  luive 


hy(l)un(l(-^), 


or 


c"^'*  =  cos  (—  n)  -f  i  sill  (—  n), 

g~Ui   —  coy  ^i  _  /  {<in  y. 

Now  multiply  this  etpuition  l)y  (^0-     »Siiice 


nUi 


)— ui 


fiUl    V     ^>"~M»    ZHZ    ('"''     X 


ui 


>ui 


=  1, 


■  eh 


Ave  nave 


or 


1  =  (cos  u)'^  —  i2  (sin  yy 
1  =  (cos  7^)2  +  (sin  ?/)2. 


It  is  cnstomary  to  write  cos^  w  and  sin^  //.  instead  of  (cos  uy 
and  (sin  uy,  to  express  tlie  sipiare  of  the  cosine  and  of  the 
sine  of  n.     The  last  e(puiti()n  will  then  be  written 

cos^  u  -\-  sin^  u  =  1.  {() 

Although  we  have  deduced  this  equation  M'ith  entire  rigor, 
it  will  be  interesting  to  test  it  by  s([uaring  the  equations  (0). 
First  squaring  cos  ti,  we  lind  (§  284), 


cos^  u  =  1  —  2C'^  +  21*  (-r-.  4- 


2!2! 


+ 


:)- 


etc. 


The  coefficient  of  w**  is  found  to  be 
1  1 


1 


+ 


7i\  '^  2!  {n  -  2)!  '^  4-1  (n-  4)!  "^  '^  n] 

wlien  n  is  double  an  even  number,  and  to  the  negative  of  this 
expression  when  w-  is  double  an  odd  number. 
Again,  taking  the  square  of  sin  ti,  we  find 


sin""^  11 
26 


^''  +  ^^'(-iT3-i-rn3-:)  +  ^^^- 


if  > 


f' 


»    I 


402 


lAfA  GINAR  Y  Q  UANTITIES. 


I 

the  coefficient  of  n^  being 

1                        1                         1 

1!  (;.'.      1)!      3!  {n      3)!      5!  {n      5)! 

•    •    •    •               J 

1 

♦ 

7i  —  i)\  : 

or  the  negative  of  this  expression,  according  as  ^n  is  even  or 
odd.  ^ 

Adding  sin''^  u  and  cos^  u,  we  see  that  the  terms  2i^  cancel 
eacli  other,  and  that  the  sum  of  the  coefficients  of  w^  can  be 
arranged  in  the  form 


4- 


4!       1!  3!   '  3!  3! 


3!  1!  "^4!* 


Let  us  call  this  sum  A.  If  we  multij)ly  all  the  terms  by 
4 ! ,  and  note  that  by  the  general  form  of  the  binomial  coeffi- 
cients, 

71  !  _  />?\ 

s\  {n  —  s)\  ~  XsJ" 
wennd         4!^  =  l-(|)+(j)-Q  +  Q 

which  sum  is  zero,  by  §  202,  Th.  II.     Therefore  the  coefficients 
of  u^  cancel  each  other. 

Taking  the  sum  of  the  coefficients  of  n^,  we  arrange  them 
in  the  form 


+ 


/         * 


o\  I  +  etc., 


n\       l\{n-iy.      2\{n~2)\      3 !  [n  -  3) 
which  call  A.     Then  multiplying  by  7i\,  we  have 

«i^=i-(i)-^(!;)-(3)+---- +(::)' 

which  sum  is  zero.  Therefore  all  tlie  coefficients  of  u^  cancel 
each  other  in  the  sum  sin^  u  -\-  cos^  u,  leaving  only  the  iii'st 
term  1  in  cos^  u,  thus  proving  the  equation  {c)  independently. 
This  example  illustrates  the  consistency  which  pervades  all 
branches  of  mathematics  when  the  reasoning  is  correct.  Tlic 
conclusion  (c)  was  reached  by  a  very  long  process,  resting  on 
many  of  the  fundamental  principles  of  Algebra  ;  and  on  reach- 


IMA  GIN  A  RT   QUA  NTITIE8. 


403 


ing  a  simple  conclusion  of  this  kind  in  such  a  wny,  the  mathe- 
matician always  likes  to  test  its  correctness  by  a  direct  i)rocess, 
wiicn  possible. 

Let  us  now  resume  the  fundamental  equation  [a),  Since 
u  nuiy  here  be  any  quantity  whatever,  let  us  put  mi  for  ic. 
TIic  equation  then  becomes, 

^md  —  cos  7iu  +  i  sin  nu. 
But  by  raising  the  equation  (a)  to  the  ??''*  power,  we  have 

gnui  —  (cos  II  -'-  i  sin  ?/)". 

Hence  we  have  the  remarkable  relation, 

{c^s  u  +  i  sin  n)"-  =  cos  7in  +  /  sin  7i2/. 

Supposing  71  zzz  2,  and  developing  the  first  member,  we 
iiave 

cos^  71  —  sin^  71  +  2/"  sin  7i  cos  h  =  cos  2u  +  /  sin  '2u. 

Equating  the  real  and  imaginary  parts  (§  327,  Cor.),  we  have 

cos'^  71  —  sin^  t(  =  cos  2?/, 

2  sin  7C  cosu  =  sin  2u, 

relations  which  can  be  verified  from  the  series  representing 
cos  u  and  sin  7(,  in  a  way  similar  to  that  by  which  we  verified 
sin^  u  -f  cos^  ic  =.  1. 

EXERCISES. 

1.  Find  the  values  of  cos^  7i,  sin^  7(,  cos'  w,  and  sin^  u  by 
the  preceding  process. 

2.  Write  the  three  equations  wdiich  we  obtain  by  putting 
■u  =  a,  7i  =  b,  and  7c  —  a  -{-  b  in  equation  (a).  Ulien  equate 
the  product  of  the  first  two  to  the  third,  and  show  that 

cos  (n  -\-  b)  =z.  cos  a  cos  b  —  sin  a  sin  b, 
sin  (a  -\-  b)  =  sin  a  cos  b  -\-  cos  a  sin  b. 

3.  Reduce  to  the  normal  form, 

{x  —  i)  {x  —  2i)  (x  —  Si)  {x  —  ii). 

4.  Develop  {a  -f  bi)^  by  the  binomial  theorem,  and  reduce 
the  result  to  the  normal  form. 


i'   • 


: , '  I' 


i^ 


404 


GEOMETIUC   liETli.  'UlSENTA  HON. 


'H 


li 


CHAPTER     11. 

THE    GEOMETRIC     REPRESENTATION 

QUANTITIES. 


OF    IMAGINARY 


o3t5.  In  Algol )ra  and  allied  branches  of  the  higher  maHn'- 
nuitics,  the  fundamental  operations  of  Arithmetic  are  extended 
and  generalized.  In  Elementary  Alge1)ra  we  have  already  hml 
several  instances  of  this  extension,  and  as  we  are  now  to  liavi' 
a  much  wider  extension  of  the  operalions  of  addition  and  mul- 
tiplication, attention  should  be  directed  to  the  principles 
involved. 

In  the  beginning  of  Algebra,  we  have  seen  the  operation  u{ 
addition,  which  in  Arithmetic  necessarily  implies  increase,  so 
used  as  to  produce  diminution. 

The  reason  of  this  is  that  Arithmetic  does  not  recognize 
negative  quantities  as  Algebra  does,  and  therefore  in  employ- 
ing the  latter  we  have  to  extend  tlie  meaning  of  addition,  so  ;i,s 
to  apply  it  to  negative  quantities.  When  thus  applied,  we 
have  seen  that  it  should  mean  to  subtract  the  quantity  which 
is  negative. 

In  its  primitive  sense,  as  u.-^ed  in  the  third  operation  of 
Arithmetic,  the  word  Dudfip/f/  means  to  add  a  quantity  to  itself 
a  certain  number  of  times.  In  this  sense,  there  would  be  no 
meaning  to  the  words  ''multinlv  1)V  a  fraction."  But  we  ox- 
tend  the  meaning  of  the  word  multiply  to  this  case  by  defining]; 
it  to  mean  taking  a  fraction  of  the  quiintity  to  be  multiiiliid. 
"We  then  find  that  the  rules  of  multiplication  will  all  apply  t" 
this  extended  operation. 

Tliis  extension  of  multipli(Mition  to  fractions  does  not  tiike 
account  of  negative  multipliers.  In  the  latter  case  we  ran 
extend  the  meaning  of  the  operation  by  providing  that  the 
algebraic  sign  of  the  (juantity  shall  be  clianged  when  the  mul- 
tiplier is  negative.  We  thus  have  a  result  for  multiplicnli  'ii 
by  every  positive  or  negative  algebraic  number. 

Now  that  we  have  to  use  imaginary  quantities  as  multi- 


GEOMETRIC   REP  RESENT  A  TJOK 


405 


lies  increase,  m 


pliers,  a  still  fnrthcT  extension  is  necessary.  Hitherto  our 
operations  with  imaginary  units  have  been  jiurely  symbolic  ; 
that  is,  we  have  used  our  symbols  and  ])erformed  our  operations 
without  assigning  any  delinitc  meaning  to  them.  We  shall 
now  assign  a  geometric  signitication  to  operations  with  imagi- 
nary units,  subject  to  these  three  necessary  conditions  : 

1.  The  operations  must  be  subject  to  the  same  rules  as 
those  of  real  ([uantities. 

2.  The  result  of  operating  with  an  imaginary  qnantily 
must  be  totally  ditTerent  from  that  of  operating  with  a  real  one, 
and  the  imaginary  f[uantity  must  signify  something  which  a 
real  quantity  does  not  take  account  of. 

3.  If  the  imaginary  quantity  changes  into  a  real  one,  the 
operation  must  change  into  the  corresponding  one  with  real 
quantities. 

834.  Geometric  Reprcseiifcdion  of  Tmafiinary  Units.  Cer- 
tain propositions  respecting  the  geometric  representation  of 
multiplication  have  been  fully  elucidated  in  Part  I,  and  arc 
now  repeated,  to  introduce  the  corresponding  representations 
of  complex  quantities. 

I.  All  real  numbers,,  positive  and  negative,  may  be  arranged 
along  a  line,  the  positive  numbers  increasing  in  one  direction, 
the  negative  ones  in  the  opposite  direction  from  a  fixed  zero 
point.  Any  number  may  then  be  represented  in  magnitude 
by  a  line  extending  from  0  to  the  place  it  occupies. 

We  call  this  line  a  Vector. 

II.  If  a  number  a  be  multiplied  by  a  positive  multiplier 
(for  simplicity,  suppose  +1),  the  direction  of  its  vector  will 
remain  unaltered.  If  it  be  multiplied  by  a  negative  multiplier 
(suppose  —1),  its  vector  will  be  turned  in  the  opposite  direc- 
tion (from  0  —  «  to  0  +  (7,  or  vice  versa).  Compare  §  T3, 
where  the  coarse  lines  are  the  vectors  of  the  several  (luantities. 

0 


—  a 


+  a 


itlties  as  multi- 


Ill.  If  the  number  be  multiplied  twice  by  —  1,  that  is,  by 
(—1)2,  its  vector  will  be  restored  to  its  first  position,  being 
twice  turned,  and  if  it  be  multiplied  twice  by  -f  1,  that  is,  by 
(-f  l)"'^,  its  vector  will  not  be  changed  at  all.     Its  vector  will 


» ' 


f 


*  >l 


40(3 


IMA  GINA  R  T   Q  UA  NT  [TIES. 


m 


-t-la 


—  a- 


—  +« 


—ia 


therefore  be  found  in  its  first  position,  whether  we  multiply  it 
by  the  square  of  a  pcsitive  or  of  a  negative  unit;  in  othei- 
words,  both  squares  are  positive. 

]V.  To  multiply  the  line  +  a  twice  by  the  imaginary  unit 
«',  is  the  same  as  niultii)lying  it  by  i^  or  —  1.     Hence, 

Midti/f/ijii/o'  hij  the  inutgbiarij  itiiit  i  must  'Jirr  the 
vector  sitch  a  Diotioii  as,  if  repeated,  will  cJiau^e  it  from 
-\-  a  to  —  a. 

Such  a  motion  is  given  by  turn- 
ing the  vector  through  a  right  angle, 
into  tlie  position  +  ia.  A  second 
motion  brings  it  to  the  position 
—  a,  the  opposite  ql  -^  a.  A  third 
motion  brings  it  to  —  ia,  a  position 
the  opposite  of  +  in.  A  fourlii 
motion  restores  it  to  tJic  original 
position  +  a. 

If  we  call  each  of  these  motions  multiplyiny  by  i,  we  have, 
from  the  diagram,  a  =.  a,  ia  —  ia,  tki  =z  —  a,  i^a  =.  —  i<t, 
i%  =  a,  which  corresponds  exactly  to  the  law  governing  the 
powers  of  i  (§  326).     Hence  : 

//  a  q-itantity  is  represented  hy  a  vector  extending 
from  a  zero  point,  the  midtiplicatioii  of  this  quantity  hi/ 
the  imaginary  unit  maybe  represented  by  turning  the 
vector  through  90°. 

V.  In  order  that  multiplier 
and  multiplicand  may  in  this  op- 
eration be  interchanged  without 
alTecting  the  product,  we  must 
suppose  that  the  vertical  line 
which  we  have  called  ia  is  the 
same  as  al,  that  is,  that  this  lino 
represents  a  imaginary  units. 

TVe  have  therefore  to  count 
the  imaginarjf  units  along  a 

vertical  line  on  the  same  system  that  wecoivnt  the  real 
units  on  a  horizontal  liihe. 


- 

+  4i 

- 

4-3i 

- 

■t-2J 

-A  -3 

-2  -1 

^M 

2     ; 

5      i 

—i 

- 

-2i 

-■ 

-3i 

_ 

—4i 

GEOMETRIC   REPRESENTA  TIOX. 


407 


(Hint  the  real 


hi 


bi 


■a-hbi 


bi 


■^^ 


a 


/■ 


-a~bi 


bi 


J>35.  Gpometric  Representation  of  a  Complex  Quantit//. 
We  have  shown  (§  15)  tliut  algebraic  addition  may  be  repres«entud 
by  putting  lines  end  to  end,  the 
zero  point  of  each  line  added  be- 
ing at  the  end  of  the  line  next 
preceding.  The  distance  of  the 
(lid  of  the  last  line  from  the  zero 
point  is  the  algebraic  snm. 

On  the  same  system,  to  repre- 
sent the  algebraic  snm  of  tlie  real 
and  imaginary  quantities  a  -\-  bi, 
we  lay  off  a  units  on  the  real  (horizontal)  line,  and  then  b 
•nits  from  the  end  of  this  line  in  a  vertienl  direction.  The 
end  of  the  vertical  line  will  then  be  tne  position  corresponding 
to  a  -\-  bi. 

It  is  evident  that  we  should  reach  the  same  point  if  we 
first  laid  off  b  units  from  0  on  the  imaginary  line,  and  then  a 
units  horizontally.     Hence  tliis  system  gives 

/;/■  -\-  a  =z  a  -\-  hi, 

as  it  ought  to,  to  represent  addition. 

If  a  or  h  is  negative,  it  is  to  be  laid  off  in  the  opposite  di- 
rection from  the  positive  one.  We  then  have  the  points  cor- 
responding to  —  «  +  bi,  —  a  —  bi,  and  a  —  bi,  shown  in  the 
diagram,  which  should  be  carefully  studied  by  the  pupil. 

The  result  we  have  reached  is  the  following : 

Every  complex  quanfifqj  a  +  bi  is  considered  as  he- 
lon^in^  to  rt  certain  point  on  the  plane,  namely,  tliab 
point  which  is  reached  by  laying  off  from  the  zero  point 
a  units  in  the  horizontal  direction  and  b  units  in  the 
vertical  direction. 


Of 


►36.  Addition  of  Com- 
plex Quantities.  If  we  have 
several  complex  terms  to 
add,  as  a  -\-  bi,  m  —  ni, 
p  +  qi,  we  may  lay  tlieni 
off  se]wrately  in  their  ap- 
propriate magnitude  and  di- 


0  - 


1 


I 


408 


IMA OINAR Y  Q  UAiYTITlL'S. 


f  :  I 


roction,    as   in   the   figure,   the   last  line    terminating    in    a 
l)()ii)t  K. 

If  we  first  add  the  quantities  a  +  bi,  etc.,  algebraieally 
(§  22-1),  the  result  will  be 

a  -|_  ,n  ^  p  ^  (7)  —  n  +  q)  i. 

AVe  may  lay  ofE  this  sum  in  one  operation.  The  sum  a-{-m 
■\-p  will  carry  iis  from  0  to  jVI,  and  the  sum  {h  —  n  +  q)  i 
from  M  to  R,  because  MR  ==  i  —  n  +  q.  Therefore  we  shall 
reach  the  same  point  R  whether  we  lay  the  quantities  off  sepa- 
rately, or  take  their  sum  and  lay  off  its  real  and  imaginary 
parts  scpjirately. 

*i*Mi,  Vectors  of  Complex  Qiiantitirs.  The  question  now 
arise:^  by  what  straiglit  line  or  vector  siiall  we  represent  a  sum 
of  C(3mplex  quantities  ?     The  answer  is : 

Tlie  vector  of  a  sum  of  sev- 
eral vectors  is  the  strai£7it  line 
from  the  h'^giunin^  of  the  first 
to  the  end  of  the  last  vector 
added. 

For  example,  the  sum  of  the 
quantities  OX  —  a  and  XP  =  hi  is  the  vector  OP. 

It  miglit  seem  to  the  student  that  the  length  of  the  vector  represent- 
ing tlic  &um  shouhl  he  equal  to  the  combhied  lengths  of  all  the  sepaiutc 
vectors.  This  difficulty  is  of  the  same  kind  as  that  encountered  hy  tlu' 
beginner  in  finding  the  sum  of  a  positive  and  negative  quantity  less  than 
cither  of  them.  The  solution  of  the  difficdty  is  simply  that  by  addition 
we  now  mean  something  difTerenc  from  both  arithmetical  and  algebraic 
addition.  But  the  operation  reduces  to  arithmetical  addition  when  the 
quantities  are  all  real  and  positive,  because  the  vectors  are  then  all  placed 
end  to  end  in  the  same  straight  line.  Therefore  there  is  no  inconsistency 
between  the  two  operations. 

Two  imaginary  quantities  are  not  equal,  u- less  botl'  their 
real  and  imaginary  parts  are  ecpial,  so  that  tiieir  sum  shall  tei-- 
minate  at  the  same  point  P.  Their  vectors  will  then  coincide 
v.ith  each  other.     Hence : 

Two  vectors  are  not  considered  equal  unless  they  a^rce 
in  direction  as  well  as  length. 


GEOMETlilC   RKPRESENTA  TION. 


40i) 


itmg    in    a 


Ijrubraiciilly 


D  sum  a  4-  /?' 
)  —  w  +  (/)  i 
ore  we  sluill 
ties  off  sepa- 
(1  imaginary 

^ucstion  now 
resent  a  sum 


&i 


X 


P. 

ector  rcproscnt- 
all  the  st'piinitc 

liuntercd  by  tlu; 
aiitity  less  tluui 
hat  by  addition 

lal  and  algebraic 

llition  when  tl-.e 
then  all  placed 
LO  inconsistency 


less  both  Hu'lr 

5um  sball  trv- 

Itlien  coincide 

\ss  they  a^ree 


111  other  words,  iih  ordrr  to  dctcrnune  a  rector  com- 
pletely, we  viust  know  Us  direction  as  'cell  as  its  (enjjt/i. 

This  result  embodies  the  theorem  of  the  i)receding'  chapter  (j^  327), 
tliat  two  cotnpU^x  (luantities  are  not  e(]ua)  unless  both  their  real  and 
imaginary  parts  are  ecjual.  It  is  only  in  case  of  this  double  ('(juality  that 
the  two  complex  quantities  will  belong  to  the  same  j)oiut  on  the  ].lano. 

Because  OXP  is  a  right  angle,  we  have  by  the  Pytbagoreati 
theorem  of  Geometry, 

(length  of  vector)'-'  =  a^  +  />'-, 


or 


length  of  vector  =:  \^a'-  -\-  /A 

We  arc  careful  to  say  long  fit  of  vector,  and  not  merely  vec- 
tor, because  the  vector  has  direction  as  well  as  length,  and  the 
direction  is  as  important  an  element  as  length. 

To  avoid  repeating  the  words  ''length  of,"  we  shall  put  a 
dash  over  the  letters  representing  a  vector  when  we  consider 
only  its  length.     Then  OX  will  mean  ioif/fh  of  the  line  OX. 

Def.  The  length  of  the  vector,  or  the  expression 
Va-  +  b'\  is  called  the  Modulus  of  the  complex  ex- 
pression a  +  bl. 

The  modulus  is  the  absolute  value  of  the  expression,  con- 
sidered without  respect  to  its  being  positive  or  negative,  real 
or  imaginary.     Thus  the  different  expressi(Uis, 

—  5,     +5,     3  -f  4/,     4  —  ?ri,    bi, 


all  have  the  modulus  5  (because  ^/'.V^  +  4^  =  5).  The  points 
which  represent  them  are  all  5  units  distant  from  the  zero 
point,  and  so  lie  on  a  circle,  and  their  vectors  arc  all  5  units  in 
length. 

The  German  mathematicians  t'^erefore  call  the  modulus 
the  ((hmlute  value  of  the  complex  quantity,  and  this  is  really 
a  better  term  than  the  English  ex])ression  modidus. 

Def.  The  Angle  of  the  vector  is  the  an.2;le  which  it 
makes  with  the  line  along  which  the  real  units  are 
measured. 

If  OA  is  this  line,  and  OB  the  vector,  the  angle  is  AOB. 


I      I 


s 


i  in 


■i 
i  ' "  'jf'l 


410 


LVA  GIN  A  RY   Q  UANTITIES. 


EXERCISES, 


Lay  off  the  following  complex  quantities,  draw  the  vectora 
corresponding  to  them,  and  find  the  modulus  both  by  measure' 
mcnt  and  calculation  : 


I. 

4  -f  3t. 

2 

4       3/. 

3- 

—  4  +  3i. 

4- 

4       3/. 

5- 

3  +  4/. 

6. 

3  -  4/. 

7- 

-  3  -H  4i". 

8. 

-  3  -  U. 

9. 

5  +  T/. 

lO. 

5  +  0/. 

II. 

5  4-  5l 

12. 

5  +  4i. 

13- 

3  +  %i. 

14. 

3  +  L 

15. 

3  -  /. 

1 6. 

3  -  'U. 

17.  Draw  a  horizontal  and  vertical  line;  mark  several 
points  on  the  plane  of  these  lines,  and  find  by  measurement 
the  complex  expressions  for  each  point.  Als<  .  draw  the  sev- 
eral vectors  and  measure  their  length.  Continue  this  exercise 
until  the  relation  between  the  complex  expressions  and  their 
points  is  well  apprehended. 

Note.  The  student  may  adopt  any  scale  he  pleases,  but  a 
scale  of  millimeters  will  be  found  convenient. 

338.  Geometric  Multiplication.  The  question  next  arises 
whether  the  results  we  obtain  for  multiplication  of  complex 
quantities  follow,  in  all  respects,  the  usual  laws  of  multiplica- 
tion, especially  the  commutative  and  distributive  laws. 

I.  To  i7inltiply  a  vector  by  a  real  factor. 

Let  the  vector  be  «  +  bi  and  the 
factor  m.     The  product  will  be 

ma  +  mhi. 

l\\  the  geometric  construction,  let 
OK -a  and  AB  =  bi.  We  shall 
then  have,  by  the  rule  of  addition, 

Vector  OB  =  «  +  bi. 

When  we  multiply  a  by  m,  let  OA'  be  the  product  ma,  and 

A'B'  the  product  mbi.     Because  the  lines  OA  and  AB  are  both 

multiplied  by  the  same  real  factor  m  to  form  OA'  and  A'B',  we 

shall  have 

OA  :  AB  :  OB  =  OA'  :  A'B'  :  OB'. 


I 


OEOMETRW  JiEPIiESEyTA  TION. 


411 


the  vectors 
3y  measure - 

t  +  3i. 
■  4?. 
-7/. 
■4i. 
-  i. 

ark  several 
icasnremciit 
•aw  the  sev- 
this  exerciso 
IS  and  their 

leases,  bat  a 

1  next  arises 
of  complex 
multipliea- 

aws. 


Bf 


.■^ 


B 


I— -A -^ 


net  ma,  and 
AB  are  both 
md  A'B',  we 


Therefore  tlie  triangles  OAH  and  OA'IV  arc  siniilai'  and 
('(^iiiangular,  so  that 

angle  A'OB'  ^  angle  '  OB. 

This  shows  that  the  linos  i)\\  and  OB'  coincide,  so  that 
I'»IV  is  the  continuation  of  OB  iu  the  same  straight  line.  More- 
over, the  above  proi)ortioii  gives 

OB'  =  w/OB, 
or,  from  (1),  vector  (*B'  =  m  vector  OB. 

Therefore.  inulti])hjim2  a  vector  hti  a  veal  factor 
vhauges  its  length  ivithont  altering  its  dirretion . 

II.  To  multiply  a  vector  hij  the  ii)iftginary  unit. 

Multiplying-       -f-  bi  by  i,  the         q 

result  is 

—  b-{-  ai. 

The  construction  of  the  two 
vectors  being  made  as  in  the  fig- 
ure, we  have 

OB  =  a  4-  hi, 
OQ  =:  -  Z*  4-  <n\ 

Because  the  triangles  OPQ  and  OAB  are  right-angled  at  P 
iuid  B,  and  have  the  sides  containing  the  right  angle  e([ual  in 
length,  they  are  identically  equal,  and 

angle  POQ  =  angle  OBA  =  00°  -  angle  BOA. 

Hence  the  sum  of  the  angles  POQ  and  BOA  is  a  right 
angle,  and  because  POA  is  a  straight  line,  therefore, 

angle  BOQ  -  9C". 

Therefore,  the  result  of  iivuUiplyiiig  the  vector  OB  by 
the  imaginary  unit  is  to  turn  it  90°  without  changing 
its  length. 

We  have  assumed  this  to  he  the  case  when  the  vector  represents  a 
real  quantity,  or  lies  along  the  line  OB  ;  we  now  see  that  the  same  thing 
holds  true  when  the  vector  represents  a  complex  quantity. 

If  instead  of  the  multiplier  being  simply  the  imaginary 
unit,  it  is  of  the  form  7ii,  then,  by  (I),  in  addition  to  turning 
the  vector  through  90°,  we  multiply  it  by  w. 


H     • 


f    :  I 

t      I 


I 


t 


'I  ■, 


'J    '  t 


412 


JM. t a/XA  Ji  Y   q  UANTl Tlh'S. 


III.  To  niiUtiplij  a  vector  hy  a  complex  quantlly, 

m  +  ni. 

This  will  consist  in  inultii)lying  separately  l)y  m  and  ;//, 
and  uddiii*^  the  two  products.     Put  OB  =  a  ■]-  di,  the  vector 
to  he  miilliplied ;  ON  = 
7n  +  Hi,  the  inulti[)licr. 

To  multiply  OB  hy  w, 
we  take  u length  OC,  deter- 
mined hy  the  proportion, 

OC  :  OB  =  m  :  1,    (I) 

whence  hy  (I), 

OC  =  m-OB 

=  m  {a  +  hi). 

To  multiply  OB  hy  ni,  we  take  a  length  CD  determined 
hy  the  condition, 

length  Q\)  =:  n  length  OB, 


or 


CD  :  OB 


n 


1; 


and  to  multiply  hy  i,  we  i)lace  it  perpendicular  to  OB.      (11) 

We  then  have, 

CD  r=  OB  X  ni. 

In  order  to  add  it  to  OC,  the  other  product,  we  place  it  as 
in  the  diagram,  and  thus  lind  a  point  D  which  corresponds  to 
the  sum 

OC  +  CD  =  OB  X  m  +  OB  x  ni ; 

that  is,  to  the  product 

{m  -\-  ni)  {a  +  hi). 

Now  because  OC  =  OB  x  m  and  CD  =  OB  x  n,  we  havo 


OC    :  C  D  =m'.  n  =  OM  :  MN, 

and  because  the  angles  at  M  and  C  are  right  angles,  the  tri- 
angles OCD  and  OMN  are  similar.     Therefore, 

angle  COD  =  angle  MON. 

Hence  the  angle  AOD  of  the  product-vector  is  equal  to  the 
sum  of  the  angles  of  the  multiplier  and  multiplicand. 
For  the  length  OD  of  the  product-vector  we  have. 


(JLVMETlilC   lihTlihSh'ATA  TlUN. 


413 


D  determined 


length  Ol)-^  =  OC^  +  Cir 

Extractinf^  the  squjiro  root, 

length  OD  =      V)^ +T' .  (Tn 

=  ^/'nl^l^n^ .  V^^  +  /A 

Therefore  the  length  of  the  prodnet-vector  is  e(|ual  to  the 
])rodnets  of  tlie  lengths  of  the  vectors  of  the  factors. 

Combining  these  two  results,  wc  reach  tho  conclusion: 

The  vwdulusnf  tlie  j)7'oditct  oftiuo  coinp/c.v  /actors  is 
equal  to  the  product  of  their  moduli. 

Tlie  an0e  of  tlie  product  is  equal  to  the  sjitti  of  the 
angles  of  the  factors. 


.'J.'JO.  The  Roots  of  Unity.  We 
have  the  following  curious  problem: 

Given,  a  vector  OA,  which  call  a; 
it  is  rerpiired  to  find  a  comj)lex  factor 
r,  such  that  when  we  multiply  d  n 
times  by  x,  the  last  product  siiall  be  a 
itself.     That  is,  we  must  have 


xHi 


a. 


A. 


The  required  factor  must  be  one 
Avhich  will  turn  the  vector  round  without  changing  its  length. 
Let  us  begin  with  the  case  of  n  =  3. 

Since  three  equal  motions  must  restore  OA  to  its  original 
position,  the  condition  will  be  satislied  by  letting  x  indicate  a 
motion  through  1:^0°,  so  that  OA  shall  take  the  position  Oli 
when  angle  AOB  =  120°.  Then,  P  being  the  foot  of  the  \k'Y- 
pendicular  Irom  B  upon  AO  produced,  we  shall  have  angle 
FOB  =  60°,  and  angle  PBO  =  30°.     Therefore, 


PO  =  ^«, 


PB  =  ^. 


and 


vector  OB  =  xa  =  —  ^^a  -(- 


V3 


!'   * 

i    I 
t 


^    ■  I 


.      ! 


•    • 


ai. 


414 


f\fA  a  IX A 11 Y  q  u.  I  NriTfES. 


'  ^^'4 


l^'Ciiusc  tlio  rii('tor.r  linn  not  C'ljim«i^('(l  lliu  Icu^'llj  of  tlu'  lino, 
the  iikhIiiIus  of  u;  is  iinilv,  mid  Ix'causo  it  lias  turned  the  line 
tliroui-li  Vi^°^  its  unfile  is  l:iO".     'riicret'ore  its  value  is 

-  OP  +  IMi/ 

on  a  scale  of  numbers  in  whieli  01>  =  1;  that  is, 

Reasoning  in  tiie  same  way  with  respect  to  the  product  x^a, 
which  produces  the  vector  OC,  we  lind 

an  equation  whicli  we  readily  ])rovo  by  squaring  the  jireceding 
value  of ./'  and  reducing. 

Multiplying  these  values  of  .r  and  ^r^,  v,e  find 

a^  =  1, 

which  ouglit  to  be  the  case,  because  :v^n  =  a.     Hence, 

1       \/'\ 
Tlic  coin])le,v  qitantity  — -  H — ^^i  is  (t  cube  root  of 

iLiiity. 

But  the  vector  OC,  of  wliich  tlie  angle  is  340°,  also  repri - 
sents  a  cube  root  of  unity,  if  we  8ui)})()se  00  =  1,  becauM' 
three  motions  of  240°  each  turn  a  vector  through  720°,  or  two 
revolutions,  and  thus  restore  it  to  its  original  position.  This 
also  agrees  with  the  algebraic  process,  because,  by  squaring  the 
above  value  of  x^,  we  have 


/     1       V3V_1       i!,^-_       z_L^_- 
V~2  ~    2    7  ~  4  ~  4  "^    o   *  -  -  o  +    o    ^  -^y 


3  V3 

4  "^    2 


1  \'^  . 

2  ■•■    2 


and  by  repeating  the  process  we  find 

Since  1  itself  is  a  cube  root  of  unity,  because  P  =  1,  we 
conclude  : 

TJiere  are  three  cube  roots  of  unity. 


'  I  : 


aKOMETlilU   lih'Plih'Sf'LyTA  770 .V. 


415 


We  ivudil}  liiul,  l)y  tlic  procfs.s  of  5«  :VM,  IV,  tluit 

/,     —  I,     —  /,     ami     J, 

aiv  nil  fourth  roots  of  unity. 

liy  ti  course  of  fi'a8oiuii«]j  similar  to  the  alxivc  for  any  value 
of  ;/,  wo  conclude  : 

The  n^^  roots  of  ii/ilh/  are  n  in  niunbcr. 

EXERCISES. 

1.  Form  the  first  eight  powers  of  the  expression 

1  ]_  .^ 

8liow  that  the  eiglith  power  is  I,  and  lay  off  the  vector  corre- 
sponding to  each  power. 

2.  Form  the  first  twelve  powers  of 

V3       1  . 

and  show  thattlie  twelftli  power  is  +1. 

3.  Find  the  fifth  and  sixtli  roots  of  unity  hy  dividing  the  cir- 
cle into  five  and  six  i)arts,  and  either  computing  or  measuring 
the  lengths  of  the  lines  which  determine  the  expression. 

Note.  The  student  will  remark  the  similarity  of  the  gen- 
eral problem  of  the  n^^  roots  of  nnity  to  that  of  dividing  the 
circle  into  71  equal  parts  (Geom.,  Book  VI). 


{    ■ 


I  1 


*id 


BOOK    XIII. 

THE     GENERAL     THEORY    OE  EQUA- 
TIONS. 


Every  Equation  has  a  Root. 

340.  In  Book  III,  equations  containing  one  unknown 
quantity  were  reduced  to  the  normal  form 

Jrc«  4-  Bx^-"^  +  Ca:«-2  _|_ _,_  ^  __  q. 

If  T^'e  divide  all  the  terms  of  this  equation  by  the  coefficient 
A^  and  put,  for  brevity, 


Vx 

B 

-  A' 

IH 

C 

~  A' 

etc. 

etc. 

Vn 

F 
-  A' 

the  equation  will  become 

X^  +  p^X^-^  -f  p^X^-^  + +  pn-lX  +  pn  =  0.         {(l) 

This  equation  is  called  the  General  Equation  of  the 
n*^'^  Degree,  because  it  is  the  form  to  which  every  algebraic 
equation  can  be  reduced  by  assigning  the  proper  values  to  ii, 
and  to  J5i,  /;< ,  p^,  etc. 

The  n  quantities  p^,  p^,  . .  .  .  pn  are  called  the  Coeffi- 
cients of  the  equation. 

We  may  consider  pn  as  the  coefficient  of  :i*  =  1. 

341.  Theorem  I.  Every  algebraic  equation  has  a  root, 
real  or  iniaguiary. 

That  is,  whatever  numbers  we  may  put  in  place  of  p^,  p^, 
Ps>  .  •  •  •  Pn,  tliero  is  always  some  expression,  real  or  imaginary, 
which,  being  substituted  for  x  in  the  equation,  will  satisfy  it. 


GENERAL    THEORY   OB'  EQUATIONS. 


417 


one  unknown 


ed  the  Coeffi- 


Rem.  The  theorem  that  every  equation  has  ii  root  is  demonstrated  in 
special  treatises  on  the  theory  of  equations,  but  the  demonsiration  is  too 
long  to  be  inserted  ht.'re. 

If  we  suppose  tlie  values  of  the  coefficients  Pi,Pc,,  etc.,  to 
vary,  the  roots  will  vary  also.     Hence, 

Theorem  II.  The  roots  of  an  (dgehraic  equation  are 
fiuictious  of  its  coefficients. 

Example.  In  Chapter  VI  we  have  shown  that  the  roots 
of  a  quadratic  equation  are  functions  of  tliu  coefficients,  because 
if  the  equation  is 


the  root  is 


X  = 


—  P  ±  Vp^—^(j 


2 


^vhich  is  a  function  of  ^;  and  q. 


343.  Equations  ivhich  can  he  solved.  If  the  degree  of  the 
eriuation  is  not  higher  than  the  fourth,  it  is  always  possible  to 
express  the  root  algebraically  as  a  function  of  the  coefficients. 

But  if  the  equation  is  of  the  fifth  or  any  higher  degree,  it 
is  not  possible  to  express  the  value  of  the  root  of  the  general 
e([uation  by  any  algebraic  formula3  whatever. 

This  important  theorem  was  first  demonstrated  by  Abel  in 
1825.  Previous  to  that  time,  jiiathematicians  frequently  at- 
tempted to  solve  the  general  equation  of  the  fifth  degree,  but 
of  course  never  succeeded. 

This  restriction  applies  only  to  the  fjcncral  equation,  in 
Avhich  the  coefficients  /?i,  p^,  p^,  etc.,  are  all  represented  ])y 
so})arate  algebraic  symbols.  Such  special  vahtes  may  be 
assigned  to  these  coefficients  that  equations  of  any  degree  slutU 
be  soluble. 

34:3.  The  problem  of  finding  a  root  of  an  equation  of  tlic 
higher  degrees  is  generally  a  very  complex  one.  If,  however, 
the  equation  has  the  roots  —  1,  0,  or  +  1,  tlicy  can  easily  be 
discovered  by  the  following  rules : 

I.  If  the  algebraic  snm  of  the  coefficients  in  the  equa- 
tion vanishes,  then  +1  is  a  root. 

211 


fi 


h    k* 


:  '1 ' 


418 


GENERAL    TIIEOliT   OF  EQUATIONS. 


II.  If  the  suin  of  the  coefficients  of  the  even  powers  of 
X  is  equal  to  tluit  of  the  coefficients  of  the  odd  powers, 
then  —  1  is  a  root. 

III.  //  the  absolute  term  p,i.  is  wanting,  then  0  is  a 
root. 

These  rules  are  readily  proved  by  jmtting  x=  +\,  then  x=—{, 
then  a;  —  0  in  the  general  equation  {a)  and  noticing  what  it  then  reduces 
to.     The  demonstration  of  II  will  be  a  good  exercise  for  the  student. 


m 


iu 


Number  of  Roots  of  General  Equation. 

344.  In  the  equation  {n),  tlie  left-band  number  is  an  en- 
tire function  of  x,  wliicb  is  equal  to  zero  wben  tbe  equation  is 
satisfied.  Instead  of  supposing  an  equation,  let  us  suppose  x 
to  be  a  variable  quantity,  wbicb  may  liave  any  value  wbatevcr, 
and  let  us  study  tbe  function  o^  x, 

.r«  -^p^x^-^  +  p^x^-''-  + +pn^ix  +^;n, 

which  for  brevity  we  may  call  Fx. 

Whatever  value  we  assign  to  x.  there  will  be  a  correspond- 
ing value  of  Fx. 

Example.     Consider  the  expre,«sion 

Fx  =  a-3  -  7cf2  +  3G. 

Let  us  suppose  x  to  have  in  succession  the  values  —  4, 
—  3,-2,  —  1,  0,  1,  2,  etc.,  and  let  us  compute  the  corre- 
sponding values  of  Fx.    "We  thus  find, 

9 

0. 


X  =  -      4,     -    3,     - 
Fx  =  —  140,     —  54, 


-    1, 

+  28, 


0, 

+  36, 

7,  S. 

+  3G,     +  100. 


X  =         1,  2,     3,  4,  5,     G, 

Fx  =1+30,     +  16,     0,     -  12,     - 14,     0, 

We  see  that  while  x  varies  from  —  4  to  +8,  the  valuo  of 
Fx  fluctuates,  being  first  negative,  then  changing  to  positive, 
then  back  to  negative  again,  and  finally  becoming  positive  once 
more. 

We  also  see  that  there  are  three  special  values  of  x.  namely, 
—  2,  +3,  and  +  G,  which  satisfy  the  equation  Fx  =  0,  and 
which  are  therefore  roots  of  this  equation. 


O  EN  Ell  A  L    TIIEOIY    OF   EQUATIONS, 


410 


theiv  0  is  a 


345.  EcpresentaliOH  of  />  />//  a  Curve.  In  i3i)()k  VIII  it 
was  shown  liow  :i  function  of  ii  variable  of  tlio  first  degree  miglit 
be  represented  to  the  eye  by  a  struight  Jine.  The  relation 
between  a  varialjle  and  any  function  of  it  nuiy  be  represented 
to  the  eye  in  the  same  way  by  a  curve,  as  shown  in  (}conietry, 
Book  VII.  We  take  a  base  line,  murk  m.  zero  point  upon  it, 
and  lay  oflf  any  number  of  equidistant  values  of  ./•.  At  each 
point  we  erect  a  perpendicular  proportional  to  the  corresponding 
value  of  Fx  at  tliat  i)oint,  and  draw  a  curve  through  the  ends. 


o;       /  The  fluctuations  of  tlie  vertical  ordinates 

of  the  curve  now  show  to  the  eye  the  corre- 
sponding fluctuations  of  Fx. 

A\'lien  Fx  is  negative,  tlie  curve  is  below 

the  base  line.     When  Fx  is  positive,  the  curve 

is  above  the  base  line. 

The  roots  of  the  equation  Fx  =:0  are  shown  by  the  points 

at  which  the  curve  crosses  the  base  line.     In  the  present  case 

these  points  are   —  2,    +3,    +  G. 

In  order  to  distinguish  the  roots  from  the  variable  (juantity 
u\  we  may  call  them  «,  (3,  y,  d,  etc.,  or  x^,  x,,,  x.^,  etc.,  or  a^, 
«3,  ^3,  etc.,  the  symbol  x  being  reserved  for  the  variable. 

The  distinction  between  x  and  the  roots  will  then  be  this: 

X  is  an  independent  variable,  which  may  have  any  value 
whatever. 

Fx  is  a  function  of  x  of  which  the  value  is  fixed  by  that  of  x. 

a,  (3,  y,  etc.,  or  x^,  x^,  x.^,  etc.,  are  special  values  of  x  which, 
being  substituted  for  x,  satisfy  the  equation 

Fx  =  0. 

Theorem.  t4ii  equation  irith,  real  coeffidents,  of  ivhieJt 
file  decree  is  an  odd  iiiunhcv,  Diust  Jtave  at  least  one  real 
root. 


^   »' 


.  I 


pi      H- 


\)»' 


>     » 


420 


GENERAL    THEORY   OF  EqUATIONS. 


Proof.  1.  When  n  is  odd,  x'^  will  have  the  same  sign  (  + 
or  — )  as  X. 

2.  So  large  a  valiio,  positive  or  negative,  may  be  assigned  to 
X  that  the  term  x'^  shall  he  greater  in  absolute  magnitude  than 
all  tile  other  terms  of  the  expression  Fx.  For,  let  us  put  the 
expression  Fx  in  the  form 


Fx 


x"^ 


\  X  X^  XV 


1) 


If  we  suppose  x  to  increase  indefinitely  either  in  the  posi- 

live  or  negative  direction,  the  terms    — ,  ^,  etc.,  will   all 

'^  X      x^ 

approach  0  as  their  limit  (§  303,  Th.  I).    Therefore  the  expression 

1  +  "7  +  ^  +  etc.  will  approach  unity  as  its  limit,  and  will 

therefore  be  positive  for  large  values  of  x,  both  positive  and 
negative.  The  whole  expression  will  then  have  the  same  sign 
as  the  factor  x'^,  and,  n  being  odd,  will  have  the  same  sign  as  x. 
3.  Therefore,  betAveen  the  value  of  x  for  wiiich  Fx  is  negative 
and  that  for  which  it  is  positive  there  must  be  some  value  of  x 
for  which  Fx  =  0,  that  is,  some  root  of  the  equation  Fx  =  U. 

For  illustration,  take  the  preceding  cubic  equation. 

Cor.     t^ln  equation  of  odd  degree  has  an  odd  nmuhcr 
of  real  roots. 

For,  as  Fx  changes  from  negative  to  positive  infinity,  it 
must  cross  zero  an  odd  number  of  times. 

846.  Theohem  I.     //  we  divide  the  expression  Fx  by 
X  —  a,  the  remainder  will  he  Fa,  or 

Remainder  =  a^  -\-  p^a^~'^  4-  p^a^'"^  4-  .  •  .  .  +  Pn- 
Special  Illustration.     Let  the  student  divide 

})y  X  —  a,  according  to  the  method  of  §  90.     He  will  find  the 
remainder  to  come  out 

a^  +  5^2  ^_  3rt  -f  1. 


GENERAL    TUEOliY   OF  EQUATIONS. 


421 


Ave  infinity,  it 


iression  Fx  by 


[e  will  find  the 


General  Proof.     When  we  divide  Fx  by  x  —  a^  let  us  put 

Q,  the  (juotieut  ; 
R,  the  remainder. 

Then,  because  the  dividend  is  ecjual  to  the  product,  Divi- 
sor X  Quotient  -f  Remainder, 

{x  -a)Q-\-  R  =1  Fx. 

Two  things  are  here  supposed: 

1.  That  this  equation  is  an  identical  one.  true  for  all  values 
of  X.  This  must  be  true,  because  we  have  made  no  supposition 
respecting  the  value  of  x. 

2.  That  we  have  carried  the  di\'ision  so  far  that  the  remain- 
der R  does  not  contain  x. 

Because  it  is  true  for  all  values  of  x,  it  will  remain  true 
Avhen  we  put  x  =  a  on  both  sides.     It  thus  reduv;es  to 

R  =  F{a), 

which  is  the  theorem  enunciated. 

The  value  of  x  being  still  unrestricted,  let  us  in  dividing 
lake  for  a  a  root  «  of  the  general  equation  Fx  =  0.  Then, 
1)y  supposing  x  =  «,  the  equation  («)  will  be  satisfied,  or 

F(c  =z  0. 

Therefore  if  we  divide  the  general  expression  Fx  hy  x  —  «, 
the  remainder  Fa  will  be  zero.     Hence. 

Theorem  II.  //  we  denote  hy  a  a  root  of  the  rqnatiorb 
Fx  =  0,  the  expression  Fx  icill  he  exacthj  divisible  hy 

X  —  fc. 

Illustration.    One  root  of  the  equation 
:^  —  x'^  —  ILi-  +  15  —  0 
is  ?).    If  we  divide  the  expression 

a^  ^  x^  —  \lx  +  15 
hy  X  —  3,  we  shall  find  the  remainder  to  be  zero. 

347.  When  we  divide  Fx  by  x  —  n,  the  highest  power  of 
X  in  the  quotient  will  be  x^~'^.  Therefore  the  quotient  will  be 
an  entire  function  of  x  of  the  degree  n  —  1. 


fe    it   , 


t  1 1 


v\ 


1    I 


422 


(iENKUAL    TllEOHY    OF   EQUATIONS. 


i  » 


I. 


lUtistraiion.    The  quotiont  from  tho  last  division  was 

X-  +  2x'  —  5, 

v.hicli  is  of  tlie  second  degree,  while  the  original  expression  was  of  the 
third  degree. 

If  ve  call  this  quotient  /\a',  wc  .shall  have,  by  multiplying 
divi    If  and  quotient, 

Fx  =  (x  —  a)  F^x. 

Kow  suppose  i3  a  root  of  the  equation 

then  F^x  will,  by  the  preceding  theorem,  be  exactly  divisible 
by  X  —  j9. 

The  quotient  from  this  division  will  be  an  entire  function 
of  X  of  the  degree  n  —  2.  This  function  may  again  be  divided 
by  X  —  y,  representing  by  y  the  root  of  the  equation  obtained 
by  putting  the  function  equal  to  zero,  and  so  on. 

The  results  of  these  successive  divisions  may  therefore  be 
expressed  in  the  form 

Fx  =  {x  —  «)  F^x  ....  (Degree  n  —  1),  \ 

F^x  z=  {x  —  (3)  F^x (Degree  n  -  2),  >■  (1) 

F<^x  =  {x  —  y)  F^x  ....  (Degree  n  —  3),  ) 
etc.  etc.      etc. 

Since  the  degree  is  diminished  by  unity  with  every  division, 
we  shall  at  length  have  a  quotient  of  the  first  degree  in  x,  of 
the  form 

x  —  e, 
f  being  a  constant. 

Then,  by  substituting  in  the  equations  (1)  for  each  func- 
tion of  x  its  value  in  the  equation  next  below,  we  shall  have 

Fx  =  {x  —  a)  (x  —  (3)  (x  —  y)  .  .  .  .  {x  —  e), 

the  number  of  factors  being  equal  to  the  degree  of  the  original 
equation.     Hence, 

Theorem  I.  Eveivj  entire  function  of  x  of  the  mh 
degree  maij  he  divided  into  n  factors,  each  of  the  first 
decree  in  x. 


NUMBER    OF  ROOTS. 


423 


Since  a  product  of  several  factors  becomes  zero  whenever 
any  of  the  factors  is  zero,  it  follows  that  the  equation 

Fx  =  0 
will  be  satisfied  by  putting  .r  r.jual  to  any  one  of  the  quantities 
(c,  (3,  y,  .  .  .  .  e,  because  in  either  case  the  jtroduct 

{x  —  «)  (.r  —  /3)  {x  —  y)....  {x  —  e) 
will  vanish.     Therefore  the  (piantitics 

are  all  roots  of  the  original  equation  Fx  =  0.     Hence, 

Theorem  II.  .In  algchvdic  equation  of  the  n^^  decree 
has  n  roots. 

We  have  seen  (§  105)  that  a  quadratic  equation  has  I  wo 
roots.  In  the  same  way,  a  cubic  equation  has  three  roots,  one 
of  the  fourth  degree  four  roots,  etc. 

Moreover,  a  product  cannot  vanish  unless  one  of  the  factors 
vanishes.     Hence  the  product 

Fx    or     {x  —  ((,)  {x  —  13)  {x  —  y)  .  .  .  .  {x  —  e) 

cannot  vanish  unless  x  is  equ.d  to  some  one  of  the  quantities, 
tc,  (3,  y,  .  .  .  .  f.     Hence, 

All  equation  of  the  n^^  decree  can  have  no  more  than 
n  roots. 

348.  We  may  form  an  equation  of  which  the  roots  shall 
be  any  given  quantities,  a,  h,  e,  etc.,  by  forming  the  product, 

{x  —  a)  {x  —  b)  {x  —  f),  etc. 

Example.    Form  an  equation  of  which  the  roots  shall  be 

—  1,     +1,     1  +  2/,    1  —  2i. 

Solution.     We  form  the  product 

[x  H-  1)  (x  —  1)  (a:  —  1  -  2i)  (x  —  1  +  2/), 

which  we  find  to  be 

x*  —  2x^  +  4:^2  +  2x  —  5. 

Therefore  the  required  equation  is 


^•4  _  0,^3  ^  4.,  2     ^  Oj. 


0. 


i' 


1 1 

i  1 


I    I 


11 

!         1 


,(,-m 


ft      < 


424 


GENERAL    THh'Onr   OF  EQUATIONS. 


(  I'l 


f' 


t-       » 


f;' 


1 . 


i-S 


EXERCISES. 

Form  pqut.tioiis  with  the  roots: 

1.  2  4-  a/'3,     '^  -  Va,     -  2,     +  1. 

2.  ;}  +  Vo,     3  —  a/5,     -  3. 

3.  o,     _  o,     4  +  y  7^     4  _  V7. 

4.  1  +  V3,     1  -  V3,     1  +  \/5,     1  -  \/5. 

849.  When  we  can  find  one  root  of  an  equation,  then,  hv 
dividing  the  c([uation  by  x  mums  tliat  root,  we  sliall  have  an 
equation  of  lower  degree,  the  roots  of  which  will  be  the  remain- 
ing roots  of  the  given  equation. 

Example.     One  root  of  the  equation 

a^  —  x^  —  Ux  +  15  =  0 
is  3.     Find  the  other  two  roots. 

Dividing  the  given  equation  by  x  —  3,  the  quotient  is 

■-c'i  -^2x  —  5. 

Equating  this  to  zero,  we  have  a  quadratic  equation  of 
which  the  roots  are 

—  1  +  VC     and     —  1  —  VG. 

Hence  the  three  roots  of  the  original  equation  arc 

3^    _  1  +  Vo,    —  1  —  Ve. 

EXERCISES. 

f .  One  root  of  the  equation 

a;3  _  3.^2  _  Ux  +  12  =  0 
is— 3.     Find  the  other  two  roots. 

2.  Find  the  five  roots  of  the  equation 

ci^  —  4:X^-{-  122;3  +  4:X^  —  ISx  —  0. 
(Compare  §  343.) 

v}50.  Equal  Roots.    Sometimes,  in  solving  an  equation, 
several  of  the  roots  may  be  identical. 
For  example,  the  equation 


X'' 


6a;2  +  Ux  —  S  z=:  0 


■  i 


(JOKFFWri'JNTS  AND    ROOTS. 


425 


has  no  root  except  2.  If  we  divide  it  by  x  —  H,  and  solve  flic 
resulting  quadratic,  its  roots  will  also  be  2.  Hence,  when  we 
factor  it  the  result  is 

(.r  -  2)  (x  -  2)  (./•  -  2)  =  0. 

In  this  case  the  equation  is  said  to  have  three  equal  roots. 
Hence,  in  general, 

I7te  n  roots  of  an  equation  of  the  n^^  dr^fee  arc  not  all 
necessarily  different  from  each  other,  hat  two  or  more  of 
them  may  be  equal. 


Relations  between  Coefficients  and  Koots. 

i^al.  Let  us  suppose  the  roots  of  the  general  equation  of 
the  n*'''  degree 

oc'^  +  2h^"'~^  +  ;>2^"~^  4-  .  .  •  .  +  Pn-\  X  4-  /)„  =  0 
to  be  «,  |3,  y,  .  .  .  .  £. 

We  have  shown  (§  341)  that  these  roots  are  functions  of 
the  coefficients  jt^,  p.^,  ....  pn-  To  find  these  functions  is  to 
solve  the  equation,  which  is  generally  a  very  difficult  i)rol)leni. 

But  the  coefficients  can  also  be  expressed  as  functions  of 
tlie  roots,  and  this  is  a  very  simi)le  process  which  wo  have 
ah'cady  performed  in  some  special  cases  by  forming  ecjuations 
luiving  given  roots  (§  348). 

If  we  form  an  equation  with  the  two  roots,  «  and  (3,  the 
result  will  be 

0  =  {x  —  (c)  {x  —  3)  =  x^  —  {a  +  ft)  X  +  fcft. 

Comparing  this  with  the  general  form, 

x'^  +  p^x  -hPz  =  0, 

we  see  that  p^  =  —  (a  -\-  ft), 

a  result  already  reached  (§§  198,  199). 

Next  form  an  equation  with  the  three  roots,  «,  ft,  y. 
^lultiplying  {x  —  «)  [x  —  ft)  by  x  —  y,  we  find  the  equa- 
tion to  be 

.r3  _  («  +  ft  +  y)  X?  +  {aft  +  /3y  +  y«)  x  -  afty  =  0. 


t   I 


1  t!      ' 


426 


GENEUAL    THEORY   OF  EQUATIONS. 


J' 


if^f 


I     i 


So  in  this  case,    p^  =  —  («  -f  /3  -f  y), 

p„  =  «^  +  /'iy  -f  y«, 

Adding  another  root  6,  we  find  the  result  to  be 

;?j  =:-(«  +  /3  +  y  -f  (5), 

;>„  =  rc/J  +  «y  +  fcrj  -|-  /'^y  +  /3(5  -f-  r^^.  (2) 

;>3  =  —  «/3y  —  ICJ36  —  tcy(]  —  fiyd, 

Pa  =  f^fh'^' 

Generalizing  this  process,  wo  reach  the  following  conclu- 
sions: 

The  coefficient  p^  of  the  second  term  of  the  general  ccjua- 
tion  is  equal  to  the  sum  of  the  roots  taken  negatively. 

The  coefficient  p^  of  the  chird  term  is  e((ual  to  the  sum  of 
the  products  of  every  comhination  of  two  roots. 

The  coefficient  /;,  of  the  fourth  term  is  equal  to  the  sum 
of  the  products  of  every  combination  of  three  roots  taken 
negatively. 

The  last  term  is  equal  to  the  continued  product  of  the  neg- 
atives of  the  roots. 

352.  Symmetric  Funcfions.  It  will  be  remarked  that  the 
preceding  expressions  for  the  coefficients  /?,,  p^,  etc.,  arc  all 
sy)nmetric  functions  of  the  roots  «,  (i,  y,  etc.     (§  250.) 

The  following  more  extended  theorem  is  true  : 

Theorem.  Every  rational  syinnnctric  function  of  the 
roots  of  an  equation,  may  be  expressed  as  a  rational 
function  of  the  coefficients. 

Example.     From  the  equations  (2)  we  find 

P.^-'lp.  =  fc3  + /:i2  +  y2 -f  (52, 

^PiP2  -  Pi^  —  ^iH  —  «'  +  (^^  +  y^  +  ^^• 

We  thus  reach  the  curious  conclusion  that  although  v>-o 
may  not  be  able  to  find  any  individual  root  of  an  equation,  yil 
there  is  no  diificulty  in  finding  the  continued  product  of  tlic 
roots,  their  sum,  the  sum  of  their  squares,  of  their  cubes,  etc. 

The  genoral  demonstration  of  tliis  tlioorcm,  and  the  methods  by  wliicli 
nny  rational  symmetrical  function  of  the  roots  may  be  determined,  arc 
found  in  more  advanced  treatises. 


DERI  VED    FUyC  TIONS. 


427 


c't  of  the  no"f- 


Dorivc'd  FiiiK'tioiis. 
353.  D(f.     If  in  the  ('Xi)ri'ssi(ju 


Fx  =  .("  +  / 


> , ./' 


/i- 1 


+   /'S' 


n-2 


+ 


-h  Pn-\  r  +  /}„, 


WO  substitute  x  +  h  for  a?,  and  tlien  deveh)p  in  powers 
of  //,  tlie  coefficient  of  tlie  first  power  of  h  is  called  tlie 
First  Derived  Function  of  x. 

To  find  the  Firs/  Derived  Function.  Putting  x  -f  //  for  x, 
the  result  is 

Developing  the  several  terms  of  the  seeond  nicniber  by  the 
binomial  theorem,  wo  have 

(x  +  /O"  =  •'■"  +  ii^^-^/f  4-  ^-^P-^  x'^-^i^  -f.  etc., 

{x  +  //)«-i  =  :>^-'^  -f  (n  —  1)  x'^-^h  4-  etc., 
{x  4-  //,)"-2  =  j;'*-2  -f  {n  —  2)  x'^-^h  4-  etc., 
etc.  etc.  etc. 

Substituting  these  expressions  in  the  equation  {a)  and 
leaving  out  the  terms  in  h^,  h%  etc.  (because  we  do  not  waul 
them),  we  have 

F{x  4-  h)  =  a;"  4-  Pi^"''^  +  Pi^""'^  + +  Pn-i  x  4-  Pn 

A-[nx*^-^-^{n  —  l)p^x^^-^  +  (n  —  T)PzX^~^-\- +Pn-\\  h 

4-  omitted  terms  multiplied  by  h^,  h^,  etc.  (b) 

We  see  that  the  first  line  is  here  the  original  Fx,  while  the 
coefficient  of  h  in  the  second  line  is  hy  definition  the  derived 
function.     So,  if  we  put 

F'x,  the  derived  function  of  Fx, 

we  have    F{x  4-  h)  =  Fx  -\-  h  F'x  4-  terms  x  //^  Ii^,  etc. 

Let  the  student,  as  an  exercise,  now  find  the  derived  function  of 

r*  +  8x3  _  5r2  +  7.C  -  9 
by  the  process  just  followed,  commencing  with  equation  (a). 

Examining  the  coefficient  of  f/  in  (/>),  we  see  that  the  de- 
rived function  is  formed  by  the  following  rule  : 


Kk 


428 


ahWKHAL    Til  1:0 It  Y   OF   hyUATJONS. 


W'  • 


I    ' 


# 


.lfiiJfi/)Jif('/frh  trvni  hij  tltr  rxpniirnf  of  the  rai'iahlr  in 
that  tcvni ,  aiut  (liniinis/t  the  rxponcnt  hij  (iiiiti/. 

Tlie  lust  or  constant  term  ilisappeiirs  entirely  from  the  ex- 
pression. 

EXERCISES. 

Form  the  derived  function  of  I  lie  following  expressions  : 

1.  a:»  +  5.r«  +  8r»  —  'Ix^  —  .r  -f  1. 

Ahs.  5.H  4-  ^0.?;3  4.  Ux^  —  4^—1. 

2.  x'  —  tr'^  —  t(^  —  'If. 

3.  :i^  +  l:>.7-»-24./-3  -|-  x^  +  7. 

4.  .r*  —  2f(:i^  4-  l]()ir:i  -j-  ((^bx. 

5.  x^'  —  5 ///./•'  4-  l^iHji^  —  Ibnij^. 

liEM.  The  student  Hliniild  nUtain  \\\o  roault  by  substituting  y  +  A  f«»r 
X  in  each  equation  und  devclopiu^^,  uniil  he  is  maater  of  the  procesH. 

854.  Second  Form  of  the  Derived  Funclion.  If,  as  be- 
fore, we  ])ut  «,  /3,  y,  (5,  etc.,  for  the  roots  of  the  equation 
Fx  =  0,  we  shall  have 

Fx  =  {x-  «)  (.7:  —  ft)  (x  -  y) (a-  -  e).  (<•) 

Let  us  form  the  derived  function  from  this  expression. 
Putting  X  -i-  h  for  x,  it  will  become 

{h  +  X  —  it)  {h  4-  .T  —  /3)  {/i  4-  .6'  —  y)  ...  .  {h  +  x  —  t:). 

Studying  this  expression,  and  forming  the  products  which 
contain  h  when  three  or  four  factors  only  are  included,  we  see 
that  the  coefficient  of  the  h  in  the  first  factor  is  {x—ft)  (x—y) 
. .  . .  ,  in  the  second  factor  {x — «)  (x — y).  . . .  ,  etc.  That  is, 
the  total  coefficient  of  h  will  be 

(x  —  ft)  {x  —  y)  .  .  .  .  {x  —  e),  omitting  first  term  ; 
4-  {x  —  a)  {x  —  y)  .  .  .  .  {x  —  e),  omitting  second  term  ; 

etc.        etc.  etc, 

4-  (x  —  «)  (x  —  ft)  (x  —  y)  .  .  .  .    omitting  last  term. 

But  comparing  with  (r),  we  see  that  the   first  of  the^e 

Fx  Fx 

products  is  — '- —  ,  the  second  is  — — 7^ ,  etc.,    to   the  last, 

^  X  —  et  X  —  ft 


which  is 


Fx 


X  —  n 


Hence, 


DEHI I  'A7>    FUSLTIONS. 


429 


Rr  Fx  Fx  Fx 

X  —  (c       X  —  ii       X  —  y  j;  — «       ^  ' 

IllHsfralion.      litt  u.s  tak(3  once  more  tlie  expression   of 

8  344, 

/It  =  j^  -  r^a  -f  :j(;, 

of  which  the  three  roots  are  —  'i,  3,  and  G.     Its  derived  func- 
tion, hy  metliod  (1),  is 

;5.r2  -  \\x. 

Expressing  Fx  as  a  product  of  factors,  it  is 
Fx  =  {x  +  2)  {x  -  :])  (./•  -  fl). 

By  {>!)  the  derived  function  is 

(X  -  IJ)  (./;  -  G)  +  (.r  +  :>)  (.;  -  G)  +  (/  +  2)  {x  -  3), 
wliich  reduces  to  3a;^  —  14:^, 

the  same  value  as  by  the  first  method. 

355.  Thi:oui;m  I.  \l  'It en  the  derived  funetinn  is  pnsi- 
tlve,  the  original  function  increases  with  x;  when  it  is 
negative,  the  function  decreases  as  x  in  creases. 

Proof.  When  we  increase  x  by  tlie  ((uantity  //,  Fx  is 
changed  to  F{x  -f-  h),  and  is  increased  by  the  difference 

F{x  ^  h)  -  Fx. 

But,  by  {h)  and  {h'),  we  have 

F{x  -\-  h)  —  Fx  =  h  F  X  +  h^  x  other  terms 

=  h  [F'x  ■\-  h  y.  other  terms).       {r) 

Now  we  may  take  the  increment  h  so  small  that  h  x  oilier 
terms  shall  be  less  than  F'x,  and  then  F'x  -i-  hx  otiier  terms 
will  have  the  same  sign  ( +  or  — )  as  F'x. 

Then,  supposing  h  positive,  the  increment 

F{x  +  h)  —  Fx 

will  be  positive  when  Fx  is  positive,  and  negative  when  it  is 
negative. 

Theorem  II.  //  an  equation  has  equal  roots,  such  root 
will  (dso  he  a  root  of  the  derived  function. 


)»    ■. 


.  II  . ! 


430 


GENERAL    THEORY  OF  EQUATIONS. 


Proof.  Let  /3  be  the  root  wliich  Fx  =  0  has  in  duplicate. 
Then  wiien  Fx  is  factored,  it  will  be  of  the  form 

Fx  =  {x  —  a)  {x  —  j3)  (x  —  13)  {x  —  y)  .  .  .  .  (x  —  e). 

Now  when  we  form  F'x  by  method  ('.*),  the  factor  {x  —  (3) 
will  be  left  in  all  the  terras.  Therefore  x  —  [3  will  be  a  factor 
of  Fx.  Therefore,  when  x  =  3,  then  F'x  =  0,  so  that  (3  is 
a  root  of  the  ecj nation  F  x  =  0. 

3<>G.  If  the  equation  Fx  =  0  contains  no  equal  roots,  and 
if  we  suppose  x  =  a  in  equation  (*■/),  all  the  terms  except  the 
first  will  vanish,  because  the  common  numerators  Fx  contain 
x  —  a  as  a  factor. 

In  the  case  of  the  first  term,  both  numerator  and  denomi- 
nator vanish  when  x  :=  a ;  therefore  we  must  find  the  limit  of 

Fx 

when  X  approaches  «.     This  is  easy,  because 


X  —  « 


Fx 


=  {X  -  [3)  {X  -  y) 


{x  —  e). 


X  —  (c 

Therefore,  by  supposing  x  to  approach  «,  wo  shall  have 

Fx 
Lim. (a;=a)  =:  («  —  (3)  («  —  y)  ....(«  —  e). 

Therefore,  by  changing  x  into  «  in  (d),  we  find 
F'cc  =z  (cc  _  /3)  (ff  _  y)  ....(«  —  e). 
Hence 

TJie  derived  function  of  a,  roof  ichich  lias  nn  other 
root  equal  to  it  is  the  continued  product  of  its  difference 
from  cdl  the  other  roots. 

Significance  ol  the  Derived  Function. 

357.  Theorem.  TJie  derived^  function  expresses  the 
rate  of  increase  of  the  function  as  compared  with  that 
of  the  variable. 

Proof.    The  equation  (c)  may  be  expressed  in  the  form 
F{x  +  //)  =  f!r  4-  h{F'x  -f  nit). 


H-\ 


FORM   OF  ROOTS. 


4;m 


where  Bh^  is  tlie  sum  of  the  remaining  terms  of  the  develop- 
ment in  powers  of  h. 

We  then  have 

Increment  of  x  =  h. 

Corresponding  increment  of  Fx  =  F{.v  -{-  h)  —  Fx 

=  h{Fx-^  Bh). 

Ilatio  of  these  increments,  — ^ ^; =  F x+  Hh, 

If  we  suppose  the  increment  h  to  approacii  zero  as  its 
hmit,  the  product  Bh  will  also  ai)proach  zero,  and  the  ratio  will 
approach  F'x  as  its  limit. 

But  this  ratio  of  the  increments  may  be  considered  a-;  tlie 
ratio  of  the  average  rate  of  increase  of  the  function  /'  to  tliat 
of  the  variable  x. 

Hence,  when  we  plot  the  values  of  Fx  by  a  curve,  as  in 
§345,  the  derived  function  shows  the  slope  of  the  curve  :it 
each  point. 

AVhen  the  derived  function  is  positive,  the  curve  is  running 
upward  in  the  positive  direction,  as  from  .Tr=— 3  to  .r  —  o, 
and  from  x  z=i  +5  to  x  z=  -f-x. 

When  the  derived  function  is  negative,  the  curve  slo])es 
downward,  as  from  a*  =  0  to  a*  =  +4. 

When  the  derived  function  is  zero,  the  curve  at  the  corre- 
sponding point  runs  parallel  to  the  base  line,  as  at  0  and  -f4j. 
If  this  point  corresponds  to  a  root  of  the  equation,  the  curve 
will  coincide  with  the  base  line  at  this  point,  and  will  there- 
fore be  tangent  to  it.     Hence,  from  §  350,  Th.  II, 

A  pair  of  equal  roots  nf  an  cqaatioii  arc  indicated  hi/ 
the  curve  touching  the  hase  line  icifJimit  intersecting  it. 


k  ' 


Forms  of  the  Roots  of  Equation. 

358.  Theorem  I.     TiiKt^i nary  mots  enter  an  equation 
with  real  coefficients  in  pairs. 

That  is,  if  a  -f-  hi  he  a  root  of  such   an   e([uation,   then 
a  —  bi  will  also  be  a  root. 


'  !•     t 


432 


GKNKRAL    TUKOUT   OF   EQUATIONS. 


Proof.     Let 

.r«  ^  y,j.,.»-i  +  ^,2.r«-2  + +  7'n-i  .r  +  />„  1=  0     (1) 

be  the  equation  with  real  cocflicieiits,  and  let  us  su])pose  tliat 
a  +  hi  is  a  root  of  this  equation.  If  we  substitute  a  ■\-  hi  for 
.r,  we  shall  have 

p^x^~^  =  p^a^"''  +  /;i(rt — l)a"~%i  —  etc. 

If  we  substitute  all  the  terms  thus  formed  in  equation  (1), 

and  collect  the  real  and  imaginary  terms  separately,  we  shall 

have  a  result 

A  -4-  Bi  :_*».) 

(§  324),  A  signifying  the  sum  of  all  the  real  terms, 


)i  ()i  —  1)       ,,  ,„ 

SI 


Pia^~\     etc.. 


and  Bi  the  sum  of  a  A  the  imaginary  ones. 

I?^  order  that  this  equation  may  be  satisfied,  we  must  have 

identically 

A  =0,     B  =  0     (§  327). 

Next  let  us  substitute  a  —  hi  for  x.  Since  the  even  povors 
of  hi  are  all  real,  and  the  odd  })owers  all  imaginary,  this 
change  of  sign  will  leave  all  the  real  terms  in  (1)  unchanged, 
but  will  change  the  signs  of  all  the  imaginary  terms,  llciuv 
the  result  of  the  substitution  will  be 

A  -  Bi. 

But  if  rt!  4-  bi  is  a  root,  then,  as  already  shown,  A  =  0 

ami  ^  =  0  ;  whence 

A  -  Bi  =  0 

also,  and  therefore  a  —  hi  is  also  a  root. 

Def.  A  pair  of  imaginary  roots  which  differ  only 
in  the  sign  of  the  coefficients  of  the  imaginary  unit  Jire 
called  a  pair  of  Conjugate  Imaginajry  Roots. 

Theorem  II.  Iti  tlic  expression  Fx  every  pair  of  conju- 
gate ima0}iary  factors  form  a  real  product  of  the  second 
deforce  in  r. 


;i ..)'•■/  . 


DECOMPOSITION   OF  RATIONAL    FRACTIONS.        433 


wc  must  have 


Proof.     If  in  the  expression 

Ft  —  [x  —  a)  {X  —  /3)  {x  —  >')...     (.r  —  f), 

we  suppose  «  and  /3  to  be  a  pair  of  conjugate  imaginary  roots, 
which  we  may  represent  in  the  form 

a  z=L  a  -\-  hi,        fi  =z  n  —  hi, 

then  the  product  of  the  terms  {x  —  a)  {x  —  h)  or  of 

{x  —  a  —  hi)  {x  —  a  •+■  hi), 

will  be  {x  —  ay  +  h'^, 

or  x^  —  2ax  +  a^  -f  h^, 

a  real  expression  of  the  second  degree  in  x. 

Cor.  Since  Fx  can  always  be  separated  into  factors  of  the 
first  degree,  either  real  or  imaginary  (§  347,  'Vh.  I),  and  since 
all  the  imaginary  factors  enter  in  pairs  of  which  the  product 
is  real,  we  conclude : 

Every  entire  functioii  of  x  icifJt  real  eoejficients  iiiny 
he  divided  into  real  factors  of  the  first  nr  second,  decree. 


'i 


■i  • 


■  '11 


he  even  povcrs 


Decomposition  of  Rational  Fractions. 

359.  D(f.  A  Rational  Fraction  is  one  which  may 
he  reduced  to  the  form 

ax"^  +  hx'n-^  4-  cr"^''^  -\-  .  .  .  .  -\-  I 

If  the  exponent  m  of  the  numerator  is  equal  to  or  greater 
than  the  exponent  n  of  the  denominator,  we  may  divide  the 
numerator  by  the  denominator,  obtaining  a  quotient,  and  a 
remainder  of  which  the  highest  exponent  will  not  exceed 
n  —  1.     If  we  put 

fx,  the  numerator  of  the  above  fraction ; 
Fx,  its  denominator ; 
Q,  the  quotient; 
(Px,  the  remainder : 


we  shall  have.     Rational  fi-action  =  •; 


fx 


Fx 


(>  + 


0:r 
Fx 


{%  96.) 


28 


A% 


434 


OENERAL    THEORY   OF  EQUATIONS, 


V     1 


» 


i   ; 


!i 


jf 


1 


Q  will  be  an  entire  liinction  of  x,  witli  which  we  need  no 

now  further  concern  ourselves. 

The  problem  now  is,  if  possil)le,  to  reduce  the  fractioi 

(bx 

-^-  to  the  sum  of  a  series  of  fractions  of  the  form 

Fx 


+ 


B 


+ 


C 


+ + 


E 


X  —  a       X  —  /3       X  —  y  x 

A,  B,  C,  etc.,  being  constants  to  be  determined,  and  a,  (3,  -) 
etc.,  being  the  roots  of  the  equation  Fx  =l  0.  Let  us  thei 
suppose 

4)X  _ 

Fx  ~ 


4_     __^ 

—  «         X  — 


E 


{!>) 


Multiplying  both  sides  by  Fx,  we  have 

AFx        BFx        CFx  EFx 

0.r  = -\ H + + 

X  —  «       X  —  p       x  —  y  X  —  e 


(^') 


We  rcfiuire  that  this  equation  shall  ])e  an  identical  oin'. 
true  for  all  values  of  .r.  Let  us  then  suppose  x  =  a.  'YAww 
because  by  hypothesis  «  is  a  root  of  the  equation  Fx  =  0.  we 
have  Fa  —  0,  and  the  terms  in  the  second  member  will  all 
vanish  except  the  first.     If  there  is  only  one  root  «,  we  lui\c' 

(§  357), 

Fx 

Lim. (.rr=oi)  ~  F'a. 

X  —  « 

Therefore,  changing  x  to  «,  we  have 

0«  =  AF'(c, 


A  =  ^-. 
F'fc 


which  gives 

In  the  same  way  we  may  lind 


B  = 


C  = 


etc. 


03 

<f>y 

F'y' 
etc. 


(<•) 


Substituting  these  values  of  A,  B,  etc.,  in  the  equation  {I), 
it  becomes 


.'^^ 


s. 


DECOMPOSITION   OF  RATIONAL    FRACTIONS.        43,") 


we  need  net 


the  fraction 


[,  and  «,  3.  ") ' 
Let  us  then 


E 


i'^} 


f 


EFx        ,j. 

X  —  £ 

identical  <>ii''. 

X  =  «.  'J'li'ii 
on  Fx  =  0.  we 
nember  will  nil 
root  «,  we  lijuo 


(^) 


[lie  equation  (i). 


03^ 


0fC 


■f 


0/3 


+ 


0y 


/'o;  ~  {x-  a)  Fa    '    (^  -  i3)  F'li   '    (.i'  -  )  )  /'  > 


+  etc. 


Note.  The  critical  student  i^liould  remark  that  in  the 
])recc(ling  analysis  we  have  not  proved  that  the  expression  of 
the  rational  fraction  in  tlie  form  (/;)  is  always  possil)le,  hut 
have  only  proved  that  if'it  be  possible,  tlicn  the  cootlicients  J, 
//,  C  must  have  the  values  {<).  To  prove  that  the  form  is 
possible,  the  second  member  of  {b)  may  be  reduced  to  a  com- 
mon denominator,  which  common  denominator  will  be  Fx, 
and  the  sum  of  the  numerators  eijuated  to  <px.  By  equating 
the  cocfHcients  of  the  separate  ]K)\vers  of  .r,  we  shall  have  n 
equations  to  determine  the  ;/  unknown  quantities  A,  B,  (\ 
etc.  Since  n  quantities  can,  in  general,  be  made  to  satisfy  n 
e([uations,  values  of  J,  i?,  C,  etc.,  will  in  general  be  possii)le. 

It  will  be  instructive  to  solve  the  following  exercises,  both 
directly  and  by  the  common  denominator. 


I.  Decompose 


EXAMPLES, 

=lx^  —  Wx  -f  5 


We  have  already  found  the  roots  of  the  'enominator  to  be 
—  2,  3,  and  G.     Using  the  formulae  (r),  we  find 

<^x  —  2x^  —  :u  +  5, 

Fx  =  .7-3  -  T.t;2  ^_  ;3(;  =  (^  -\- 2)  {x  -  3)  {x  -  G), 
F'x  =z  3.^2  —  14?: ; 

«  =  -  2,  ft  =  S,  r  =  G ; 

0«  =  19,  0/3  =  14,  0y  =:  oO; 

F'fc  =  40,  F'f3  =   -  15,      F'y  ^  24. 


22-2  —  3x  -\-  5 


10 


14 


•A\   "^  •>. 


50 


x^  —  7;i-'^  +  30        40  {x  +  2)       15  (r  -  3)  ^  24  (.r  —  r.) 

2x^  —  :.r  +  :}  2.12  _  ;.,.  ^  ;> 


2.  Decompose    -^ 


:C^  _  Ojfi  _  .^;  +  :>  (^-4-  1)  (.<;_  1 )  (.(•  —  2) 

Here  the  roots  of  the  denominator  are  —  1,  1,  and  2.     Let 
us  effect  the  decomposition  by  the  following  method.     Assume 


*  tl 


»    » 


•   >  I 

.1  ! 


Ill 


43C) 


GENERAL    THEORY   OF   EQUATIONS. 


t 

1 

/ 

2x^  -  7.r  +  3 


B  C 


{x  +  l){x-l){x-2)~  x  +  1'^  x-l'^  X  -2'    ^"'^ 

Reducing  the  second  member  to  a  common  denominator, 
it  becomes 

A  {x^  —  3.1-  +  2)  +  n{x'^  —  x—  2)_+  C{x^  —  1) 
{x  +  1)  [x  —  1)  {x  —  2)  ^ 

Since  both  members  now  have  the  same  denominator,  tluir 
numerators  nrast  also  be  e({nal.  Equating  them,  after  arrang- 
ing the  hist  one  according  to  powers  of  x,  we  have 

(A  +  B-\-C)x^-  {^A  +  B)x+  2A-9J3—C  —  2xJ^ —  ^Ix  ^  ;5. 

Since  this  must  be  true  for  all  values  of  x,  we  equate  the 
coefficients  of  a:  in  each  member,  giving 

A  +  B  +  C  =  2, 

3.1    +    h'  r=   7, 

2A  —'IB  —  C  —  3. 

These  equations  being  solved  give 

^  =  2,        B  z=z\,        C  -  —\. 
Substituiing  in  {d). 


0.^.2  _  7:c  +  3 


2 


_1 


(.,;   +    1)   [x.  —  1)  {X  —  2)  .T  +   1  X  —  \ 


X 


2 


i  I  >p,> 


Decompose : 
.r  +  10 


EXERCISES. 


I. 


.t2  — 4 

2x^  —  12a:^  -  a-c  +  12 


2. 


.r2  -j_  Sx  4-  4 


a-^  -I-  2-2  —  4.^. 

X 


X*  —  5a;2  -f  4 


X 


■'i 


a^ 


5. 


2« 


X* 


a' 


6. 


rt2^,2 


(a;^  -  a2)  (a^J  _  j2) 


3(50.  When  the  equation  ^r  =  0  has  two  or  more  eqn-il 
roots,  the  preceding  form  fails,  because  all  the  terms  of  tlif 
second  mem))er  of  (b')  will  then  vanish  when  we  suppose  .c 
equal  to  one  of  the  multiple  roots.  In  this  case  we  must  pro- 
ceed as  follows  : 


DECOMPOSITION   OF  liATIOXAl.    FRACTIONS.        4137 


{'^) 


c 

denominator, 

minator,  tluir 
,  after  arran«i- 

we  equate  the 


X  -  2' 


3 


&2) 

or  more  cqiv\l 

le  terms  of  thf 

In  Ave  suppose  / 

se  we  must  pro- 


If 
we  suppose 

0.r  A 

Fx 


Fx  =  (.r  —  (c)m  (r  _  f^yi  (.,.  _  y^p^ 


c—  ayn-i-^ 


= 1_    _  -      '  I 

(,,•  _  (tyn  ^  (,,.  _  f^yn    1  ^     (  , 

J_    ^L    ^    4. i^l      _  4.  4.     ^>'*   1 

^  (.i-  -  /3)"  ^  (.r  -  ^)'-i  ^  •  •  •  •  "^  .^  ._  /j 


4- 


hn    1 


+  (.c  -  y  V>  -^  {X  -  r>~-i  +  ••••  + 


.^-  -  y* 
etc.  etc.  etc. 

In  tlie  case  of  m,  n,  or  p  =  \,  this  form  will  be  the  same 
as  (i),  as  it  should. 

By  reducing  the  second  meinber  to  a  common  denominator, 
and  equating  the  sum  of  the  numerators  to  (px,  we  shall  have, 
as  before,  a  number  of  equations  the  same  as  the  degree  of  x 
ill  Fx. 

EXAMPLE. 


Decompose 


S.7-3  —  n.r2  —  2.r  —  1 


of  which  the  roots  of  the  denominator  are   —  1,  —  1,  I,  1,  'i. 

Sohdion.     Because  of  the  roots  just  given,  the  expression 
to  which  the  fraction  is  to  be  equal  is 

A  A.  B  B,  r/ 


A^        B_  B, 

(^X  —  1)2  "^  ;c  _  1   "^  {X  +  ly  '^  X  +   X'^X 


Reducing  to  a  common  denominator,  and  e(|iiating  the  co- 
efficients of  the  powers  of  x  to  the  coefHcients  of  the  corre- 
sponding powers  in  the  numerator  8.*:^  —  4:X^  —  2x—],  we 
have 

A,  +  B,  -f  ('=       0,  , 

—  A^  +  .4  -  3/>\  ^  B  :=:       8, 

-  3Ai  +  /?!  -  47i  -  -2^  =  -  9, 

A^  -;]A  +  :/?!  +  ')f^  =  -2, 

2A^  —  2  A  +  2/i,  +  2B  +  C  =  —I. 

Solving  these  equations,  we  find, 

A    =       1,  B    =      2,  0=3. 

A^  =z  -2,  /;,  =  -  1. 


V 


I*'.. 


»    > 


438 


GI'JNERAL    THEORY   OF  EQUATIONS. 


Tlic  fjivcii  fmc'tioii  is  tliL'i'cfore  e([ual  to 


1 


% 


+ 


a 


'   +  •' 


(.7;  _  1)2       X  —  1    '    (.r  +  1)2       a:  +  1    '   /;  -  3 


EXERCISES, 


,.  Decompose  ^^~^^^^^^ 
.r-  1 


1 


J?/.v.     — ^  4- 

3-   ;;5Tr:t;2-^^"i 


^  -{-  X^  —  X  —  I 


=1. 


Greatest  Coiiunon  Divisor  of  Two  Functions. 

,^(>1.  When  ',ve  have  two  cHiuations,  some  values  of  tlir 
unknown  quantity  may  satisfy  them  l)otli.  1'liey  are  tlieii  said 
to  have  one  or  more  cimimon  roots.  Sueli  e(iuati()ns,  wlicii 
factored  as  in  §  JU7,  will  ha' e  a  common  factor  or  divisor  for 
each  common  root.     Hence, 

Theorem.  TJie  connnoii  -"onts  of  two  equations  maij 
be  fouled  from  their  greatest  coninioii  divisor. 

Problem.  To  find  the  greatest  coimnon  divisor  of  two 
equations. 

Tliis  problem  is  solved  ]>y  dividing  the  two  polynomials  hv 
the  methods  of  ^§  96,  07,  and  'i?yl. 

Example  i.    To  find  the  greatest  common  divisor  of  the 
two  polynomials, 

and  or*  —  2x^  -\-  A:X?  +  2:r  —  5. 

FIRST   DIVISION. 

x^  —  ^x^  +  12.^  +    4:r2  —  133*  :  x^  —  2a;»  +  ix"^  +  2.r  —  5 
.^5  —  2:r>  +    4.r3  +    2./-2  —    bx  |  a;  —  2 

—  2.c»  +    8.c3  +    2a;2  _    Sx 

Zl'^1  -1-    ^^(^^  —    8.r=^  —    4:g  +  10 

4.^3  -|.  io.r2  _    4a;  _  :;.o  =  first  remainder. 


aiit'ATKST  COMMON   DIVISOR. 


4;iu 


Functions. 


BECONU   DIVISION. 

ar»  —  2.r3  +    4a;2  -f  'Ix  —  5     4./-='  -\-  10^'  —  4./-  —  10 


ot^  -\-  Iji^  —      j'i  —  L, 


—  H.r'  +    5.^2  -\-  U  —  T) 

—  5/  =  second  reiuu'ni'er; 


04!L.^•3 


or, 


*/(.?'2  —  1)  =  second  remainder. 


In  the  next  division,  we  may  omit  the  fractional  factor  ^-, 
l)tcause  every  value  of  x  which  satistics  the  C(iuation  x^ — 1  =  0 
will  also  make  ^^  {x^ — 1)  =  0,  so  that  these  two  equations 
have  the  same  roots.  In  this  process  we  may  always  multiply 
or  divide  the  terms  of  each  re?  'inder  by  any  factor  which  will 
make  their  coefficients  entire. 


divisor  of  two 


4.r^  +  lOx'2    - 

4./'3 

.  .   1 
-4:?; 

♦'ISION. 

-10 

10:r2 
10a;2 

10 
-  10 

0 

0 

a;2  -  1 


^x  +  10 


Hence,  the  G.C.D.   of  the   two  functions  is  x^  —  1,  and 
their  common  roots  are   +1  and  —1. 

This  result  may  also  be  reached  by  factoring  the  given 
tMluations,  and  multiplying  the  common  factors,  thus: 

.r=  —  4:X*  +  12.r3  +  4.t2  ~  13x 

=  X  {x  -  1)  (x  +  1)  (jc  -  2  -  30  {x  -  2  4-  3/), 

xi  -2x^-\-  4x'  +  2x  —  5 

=  {x  —  1)  (x  +  1)  (a;  -  1  —  20  (^  -  1  +  20. 

We  see  that  the  common  factors  are 

{x-l){x-}-  1)  =  2-2-1. 


440 


(HENKllAL    THEORY    OF   EQUATIONS. 


I       • 


The  rules  for  throwing  out  factors  from  divisor  or  dividend 
are  as  follows: 

I.  /y'  hot/i  o'iirit  pol unom'utls  vontdin  the  same  fdctor 
ill  all.  i/icir  hrius.  rcDwre  /his  factor,  (fud  (ij'lrr  llic 
(I.  (\  I),  of  tlif  rciriainin^  factors  of  the  two  polijnoniials 
is  foand,  niulti])hi  it  lnj  this  factor. 

Proof.  If  a  be  sucii  a  factor,  and  X  and  }'  the  quotients 
after  tills  factor  is  removed  from  the  two  polynomials,  the  lat- 
ter, as  given,  will  be 

aX    and     a  Y. 

Since  a  is  now  a  common  divisor  of  both  given  polynomials, 
if  wc  call  D  the  G.C.  D.  of  Xand  )',  it  is  evident  that  aD  will 
be  the  G.C.D.  of  aX  and  aY. 

II.  Any  factor  connnmi  to  all  the  terms  of  ami  rliri- 
,sor,  and  not  contained  in  tit e  dividend,  may  he  thrown 
out. 

Proof.  If  this  factor  were  any  part  of  the  G.C. D.  sought. 
it  would,  by  §  ^'32,  be  a  factor  of  each  dividend.  Since  the 
only  factors  we  require  are  those  of  the  G.C.D,  factors  in  a 
divisor  only  may  be  rejected. 

EXERCISES. 

Find  the  G.C.D.  of  the  following  polynomials: 

1.  x^  —  1  and  x^  —  1. 

2.  x^  —  I  and  .r'  —  I. 

3.  a^  _  0^^  _  a^  +  'dd^  —  2rt!— 15  and  a^-  a^— 4«2_^^  +  5. 

4.  25:/"*  4-  5.^3  —  .T  —  1  and  SOa:^  4.  ^-^  —  1. 

5.  a^  -V  2rt2  +  9  and  «*  +  "la^  —  iui  —  9. 

6.  m^  -i-  Sy/i^  +  3m  -|-  1  and  ;/?^  —  1. 

7.  x^  —  ar3  +  21.1-2  _  oQ.r  +  4  and  2.r3  —  12.r2  +  21.r-10. 

8.  a^  -f  a^  —  a  —  1  and  (V  +  a'^  —  a  —  \. 

*MVZ,  The  giveu  polynomials  may  be  functions  of  two 
or  more  symbols,  as  in  §  97.  We  then  arrange  them  aeeonl- 
ing  to  the  powers  of  one  of  the  symbols,  and  perform  the  divi- 
sions by  the  precepts  of  §  97. 


UHEATluST  COMMON   UlVJ^Oli. 


4U 


11  polynomials, 
it  that  aD  will 


fi.C.D.  sought. 
1(1.     Since  tlio 
factors  in  n 


Kx.     Finil  tlic  j^ri'iitest  common  divisor  of 

j.r^  —  ax'  -j-  a{d  -{-  c)  x  —  abc  —  bx^  —  r/^  -f  brx 
and        r'  —  ttx"'  —  a  (/>  -f  f)  .c  —  aiw  +  /*/-  -f  r.^'-  +  Ocx. 

Tiio  (juoticnt  of  tiio  lirst  division  will  ho  unity,  so  wv  write 
lilt*  two  functions  under  each  other,  thus: 

x^  —       {f(  -\-  b  -\-  r)  x'^  -\-  {(lb  +  be  -f  c(()  x  —  nbr 
Q^  -\-  {—  (I  +  b  -\-  r)  x'^  —  {(lb  —  be  -\-  I'd)  x  —  (the 

—  2  (/;  +  c)  x^  4-        2  (ad  4-  ac)  x  —  1st  rem. 

Dividing  this  remainder  by  —  v'  {b  4-  c),  we  h;ive  the  next 
divisor.     We  then  perform  the  next  division  as  follows: 

x^  4-  {  —  ((-[-b-\-(')  x^  —  {(ib—bc-\-ca)  x  —  nbc  i  x^  —  ax 


nx^ 


x  4-  {b  +  r) 


{b-\-<-)  /^  —  {lib  — be -{-('(()  X  —  (tbc 
{bJ^(^x-  —  {(lb         4-m).r 


hex  —  abc  =  2d  rem. 
Dividing  this  hv  the  factor  be,  whicli  is  eontained  in  all  its 


terms,  we  have  ./; 


a  for  the  next  divisor,  which  we  iind  to 
divide  the  lust  divisor,  and  therefore  to  be  the  G.C.D.  rc(iuired. 


'■   I 


EX.ERC1SES, 


^3_4rt2_^,-^5. 


12a:2  4-21:r-10. 


nctions   of  two 
re  them  aveord- 


Find  the  G.C.D.  of 

1 .  .r3  4-  Ucx  4-  b^  —  6-3  and  x^  -^{c  —  b)  x^ -]- {IP -\- be  +  r^)  x. 

2.  a'^  +  :i(ix  4-  (v^  —  I  and  x^  —  («2  _  2a)  x  -I-  a  —  1. 

3.  {a-\-b-\-e)  {ab -{- be -{- ea)  —  abe  and  a-  +  ab  —  (le  —  be, 

4.  .r*  4-  4rr»  and  x^  —  2«'<r  -f  4^/3. 

5.  x^  —  ax^  —  l)^x  4-  (lU^  and  .'■'-'  —  a^. 

6.  a:3  _^  ^^3  4_  ^»3  _  3^^^^.  ^m]  .,;;!  _|_  o^,,^  ^  ^^a  _  ^,2. 


7.     x^  —  2a:2  _^  2 


0(^ 


4- 


1 


md 


2  r'  4-       — 


8.     x^  —  x^y  4-  xf  —  //  and  .j:^  +  x^if  4-  .V^ 


A\2 


aKNHUAL    'n/hOJiV    ffF   Ni^UATlONS. 


Y 


<  > , 


TriiiisCoi'iiiiitioii  of  l]<iiiiin(>iiH. 

*MV,\.  Drf.  An  ('(lujilion  is  said  to  be  Transformed 
wIkmi  ji  second  ('(lualion  is  Toiind  wliosc  I'ools  bear  a 
known  relation  to  those  of  tiie  given  equation. 

Tki:M.  Soinolimcs  wo  may  be  iiblo  to  find  a  root  of  tlit 
traiisforinod  (.(iuatioii,  {iiul  tiieiu'e  the  c'orresj)()ndin<,'  root  of 
the  original  equation,  more  easily  tlmn  by  a  direct  sohitioii. 

Problem  I.  To  change  the  signs  of  all  thr  roots  of  an. 
equation. 

Solution.  By  changing  x  into  —x  in  a  given  equation, 
the  signs  of  the  terms  containing  odd  ])o\vers  of  x  will  bo 
changed,  wliile  those  of  the  even  powers  will  be  unchangcl. 
Hence,  if  «  be  any  root  of  the  original  C(iuation,  —  «  will  !.• 
u  root  of  the  e<|Uatioii  after  the  signs  of  the  alternate  terms  arc 
changed.     Hence  the  rule: 

Ciutnge  the  signs  of  the  (dtenutte  te?'nis,  of  odd  and 
even  degree,  in  the  e</iuitioii. 

PnonLKM  TI.  To  diminish  all  the  roots  of  an  eqim- 
tion  In  I  the  same  quatititj/  It. 

Solution.     If  the  given  e([uation  is 

xn  +  ;,^.cn-l  +  p^xn-2  _^  .  .  .  .  +  y,„  =  Q, 

and  if//  is  the  unknown  quantity  of  the  required  equation,  wr 

must  have 

y  =:  X  —  h. 

Therefore,  x  =i  ij  ■\-  h. 

Substituting  this  value  of  a;  in  the  equation,  it  will  becoii 


r+(;^i+^'^O.y""'+ 


^,„4.(;,_l);j^7, 4-^7,2  |^«-2  +  etc.  [a) 


When  //,  n,  and  the  //s  are  all  given  quantities,  the  coeffi- 
cients of  y  become  known  quantities. 


OENEllAL    TJl/'JOJiV   OJ'  HqUATlONS. 


443 


ransformed 


he  roofs  of  an 


s  of  an  cqii'i- 


E  y  E  RCI  S  ES. 

1.  'rninsiorm  tlic  CMHuition  /•  —  ;{,'• —  1  =  0  into  ouo  in 
wliiclj  tilt'  routs  .slijill  Ik'  I^'s-^  liy  1. 

2.  'rransroriu  ./•'  —  .')/-'  +  0^7*  —7  =  0  ijito  oiio  in  wWu-h 
tlu!  roots  sliall  bo  ^'rcater  by  5. 

IHii,  Ju'fuoriiuj  Tc.nns  from  Equdtlons.  Tbf  iiuanlil;.  // 
may  bo  so  uiioson  that  any  recjuirod  term  after  tlio  lirst  in  tlio 
transtbrnicd  ^filiation  slmil  vanish.  i\»r,  il"  wo  wish  the  second 
term  of  the  eciuation  {(i)  to  vanish,  wo  have  U)  suppose 

p^  +  nh  =  0, 


which  orivcs 


_       /'t 


n 


\Vc  then  substitute  tliis  vahie  of  //  in  tlio  ofpiation  (n), 
whicli  gives  an  e((uati()n  in  wiiicii  ti»e  seco.^d  term  is  wanting. 

If  wo  wish  tiie  tiiird  term  to  vanish,  wo  must  determine  h 
by  the  eoaditiou 

Qh-i^{n-l)p,Ji-^p,  =0, 

whicli  requires  the  solution  of  a  (juadrntie  equation.  Eacli 
consieutivc  term  is  one  degree  iiigher  in  tiie  unknown  (|U!in- 
titv  h,  and  the  last  term  is  of  tlio  same  decree  as  the  oricfiiiul 
equation. 

This  method  is  principally  applied  to  make  the  second 
term  disa])])ear,  which  re([uires  that  we  put 

n 

Example.  Make  the  second  term  disa])pear  fnmi  the  fol- 
lowing equation, 

x^  +  px  -f-  r/  =  0. 

Solution.    Hence,  n  =2  and  y>,  =^ p,  so  that 


h 

V 

"^' 

p 

a: 

— 

U 

2 

t     k  ' 


k 


444  GENERAL    THEORY    OF  EQUATIONS. 

Making  this  suljstitution,  the  f<{Uutioii  becomes 

wliich  is  the  required  e(tuati()ii. 

Hk.m,  Tliis  process  aifords  tin  additiouul  elegant  metbod  oC 
S(dving  tbe  ({uadrutic  equation. 
Tlie  hist  equation  gives 


II 


1 


4  ~  '^  "^  2  ^^^  ~  ^^' 


The  vaUie  of  x,  being  eciual  to  y  -\-  h,  then  becomes 
which  is  the  correct  solution. 

EXERCISES. 

Remove  the  second  term  from  the  following  equations  : 

1.  x^—  0.1-2  ^  cw  —  1   =:  0. 

2.  .74  _  4^^;5  ^.  ;3,,.2  _  8  =  0. 

3.  x^  —   ir.ir^  +  22-3  +  2x^  —  ;}.6-  =  0. 

4.  x^  —  U>;f5  -\-  -Zx^  —  X  =  0. 

Rr:M.  ^rbe  theory  of  the  above  process  will  be  readily  com- 
prehended by  recalling  that  the  coetheients  of  tbe  second  term 
is  e([ual  to  the  sum  of  tlie  roots  taken  negatively,  or  if  <(,  /i,  y, 
etc.,  be  the  roots, 

«  +  /3  +  y  +  .  .  .  .  +  e  =  —  />!■ 
It  is  evident  that  if  we  subtract  the  arithmetical  mean  of 

all  the  roots,  that  is,  —  -^ ,  from  each  of  them,  their  sum  will 
vanish,  because 


n 


n  n  n  *  n 


(>. 


ITence,  when  we  put  ?/  —  —  for  x  in  the  equation,  thetium 
of  the  roots,  and  therefore  the  second  term,  vanish. 


GENERAL    Til  HO  II Y   OF   K(^UAT10^8. 


445 


it  nietliocl  of 


)e  readily  com- 

e  second  term 

y,  or  if  «,  ^'  )• 


iieticul  lueiiu  uf 
,  their  sum  will 


i}G5.  Pkohlk.m.  To  trail sj\)fm  f/iici/iifi/ i(jn  so  t/uit  t/ie 
foots  sluill  he  iiiiiIti/)liL'd  bij  (C  jjircii  factor  m. 

Solution.  Since  the  roots  iir(  to  be  niiiltiplied  hy  ///,  the 
lu'W  unknown  (lunntily  must  \iv  e(juul  to  iiix.  So  if  \\v  cull 
this  (|u;mtity  (/,  we  luive 


w  liich  ffives 


y  =  ill. 
V 


in 


Suhstituting  this  in  the  general  equation,  it  becomes 

/«-l  /j/71-2 


yn  yn-i  yr 


0. 


Multiplying  all  the  terms  by  tn",  the  e(iuati()n  becomes 

y"  +  iiip^y"-^  4-  />'~/>oV/'*-~'  + +  ni"/h,  =  0. 

lleuce  the  rule, 

MuUiplii  tJir  rnrjficieiit  of  the  srcoiid  trrni  hi/  in,  that 
f)f  t/is  third  till  n>'\  and  so  on  to  tJir  last  trrni.  irhirli  irill 
hr  niiilfipiird  liij  m". 

If  the  roots  are  to  be  divided,  we  divide  the  terms  in  tlie 
liame  order. 


EXERCISES. 


1.  Make  the  roots  of  x^  —  'Z.v  -f  3  ==  0  four  times  as  great. 

2.  Divide  the  same  roots  by  2. 

3G6.  Piu)bli:m.     To  transform  an  equation  so  that  its 
•uots  shalt  be  sqiiftred. 

Solution.    Let  the  given  equation  be 

If  y  be  the  unknown  ([uantity  of  the  new  equation,  we 
iiii-t  have 


which  LHves 


x=  ±  yK 


If  we  substitute  x  =:  y'-^  in  the  given  e<[ualion,  it  nuiy  be 
rodueed  to  the  form 

y^  +  p^z!/  +  iu  -I-  iPii/  +  p^)  y'  =  ^- 


440 


GENERAL    THEORY  OF  EQUATIONS. 


If  we  substitute  x  —  —  i/'',  tlie  ivsult  will  bo 

f  +  />2!/  +  JU  -  {Pill  -i-  Ih) !/'  =  ^' 
Since  ilk'  v.-iliic  (»l'  //  must  satisfy  one  or  the  oilier  of  tlic^c 
o(|uatioiis,  it  mu-i  reduce  llieir  product  to  zero;  we  therefore 
multiply  tliein   together.     Considering  tliem  as  the  sum  ami 
ditrereiice  of  a  pair  of  expressions,  tiie  product  will  bo 


{f+ihv  +  /^4)''  -  {PxU  ■\-  IhYy  =  ^y 


or 


y^-i-{''Pi-Pi'')f+{P2^  +  -PA--PiP^)!f-i-(:'P2Pi-P3')!/-^P 


—  0. 


EXERCISE 


IVansform  the  (piadratic, 


x^ 


5x  -f  G. 


of  which  the  roots  are  2  and  3,  i   to  m)  equation  in  which  tlu 
roots  shall  be  the  s([uare8  of  2  an'  3,  using  the  above  process. 


2.  Transform  in  the  same  w; 


ly 


a^  +  VZx^  +  Ux  +  48 


0. 


foi 


ransiorm 


aP  —  ix^—  10a;3  -f  40xi  -f  Ox  —  3G  =  0. 


Geiioralizatioii  of  the  Preceding  Problems. 

367.  I'uoiJLKM.  Given,  an  equate jii  of  any  (lc<Ji''r 
in  an  unknown  </H,((nfit[/  x  ; 

Required,  to  transform  this  ii/itation  into  anotltcr  of 
which  the  root  shall  he  a  ^iirn  fiuic^ion  of  x. 

Sdhtlinu.  Let  y  bo  a  root  of  the  required  equation,  and  0 
the  given  function.     We  must  then  have 

f'^  =  !/• 

Solve  this  equal  ion  so  as  to  obtain  .r  as  a  function  (^f //. 
Substitute  this  value  of  •"  i  the  original  ef^uation,  and  form  a? 
many  ecjuations  as  there  are  values  of//. 

The  ])roduct  of  these  equations  will  be  the  required  e<|n!i- 
tion  in  y. 


rs. 


GENERAL    TUEORY   OF  EQUATIONS. 


447 


0. 

olbcr  of  tlu'f>c! 

wo  tlierefoiv 

the  sum  iind 

■ill  be 

:0, 

—  0. 


n  in  AvbicU  tlu" 
above  process. 


=  0. 

Problems. 

/  any  dc^nc 

ito  another  of 

X. 

([nation,  and  '■' 


function  of  //. 
ion,  and  form  a.> 

re(iuirecl  etinti- 


EXERCISES. 

1.  Transform 

a,-2  _  7x  +  10  =  0 

so  that  the  roots  of  the  new  e([uation  shall  be  ox^. 

2.  Transform        x^  —  3.1'-  -f  '^x  =z  0 
.<o  that  the  roots  shall  be  ax  +  b. 

3.  Transform         x'^  —  0./;  +  18  =  0 

so  that  the  roots  shall  be  - -^'^  —  3. 

o 

Ilesoliitioii  of  Niiinorieal  Equations. 

3G8.  Gouvniient  ncfhod  of  voniputiixj  (he  nioncriad  rahic 
of  an  entire  function  of  x  for  an  assumed  value  of  x. 

If  wo  have  the  entire  function  of  .r, 

Fx  —  ax^  +  Ijx^  +  ex'  +  dx  +  c, 
■we  may  put  it  in  the  form 

Fx  =  \[{ax  +  h)x  +  c]  X  -^  d\x  -^  e. 
Therefore,  if  we  put 

ax  +  />  =  //,  h'x  -\-  c  z=:  r\ 

c'x  -\-  d  :=.  d',  d'x  -\-  e  z=z  e', 

wc  shall  have  Fx  —  e'. 

Numerical  Example.     Compute  the  values  of 
Fx  =  'ZqP  —  a^-t  —  (Jx^  +  Sx  —  9 
f(jr  X  =  3  and  x  =  —  '^. 

We  arrange  the  work  thus : 

.3         -G  0 

^-G         +9         -fO 

T3         +3         +9 
Fl]  =  00. 

2        -3      -  (;  0 

+  U      -10 

-I- Is      —10 

F{-'i)  =^  -89. 


CociHcients,  2 

Prod,  by  (:^-=3), 


Ilenco, 

Fur  X  =  -2, 

Hence, 


■3 
■4 


+  8 
+  35 


+   8 
+  32 

+  40 


—     0 
+ 105 

+  ~T)o 


0 
80 

89 


448 


QKNKRAL    THEORY   OF   EQUATIOXS. 


]    I 


r»      I 


i . 


Tliis,  it  will   b"  noticed,  is  ii  wxovo  convenient  proceHS  than  that  of 
Ibrmini;  th(!  i)o\ver.->  of  x  and  nuiltijilying  and  adding. 

309.  11(11  inrj  an  oil  ire  function  of  x,((nd  puffinfj  x  —  r-\.li^ 
it  IS  nqdircd  to  ik'vdup  the  function  in  jviwers  of  ft. 

It  will  he  remarked  that  tins  proijlem  is  Hub.stantially  identical  with 
tliat  of  ?'  'M')2,  and  iL  •  sohition  >>*'  this  will  be  the  holutior.  of  the  foimcr. 
lint  in  tl.e  former  cane  h  was  su  )|)OHed  to  be  a  given  (juantity,  whereas  ii 
is  now  the  unknown  »iuantity  cjrreisponding  to  y  in  the  former  problt-ni. 

Example  of  the  1  kohlkm.     IT  wc  luivc  the  expression 

Fx  =  -Ir^  +  ;j^;2  +  4, 

and  ))iit  X  —  ;2  +  //,  it  will  become,  by  (icv eloping  the  sepa- 
rate terms, 

F{'l  +  h)  =  '2P  +  1:)//=^  -f  dUh  +  32. 

(rllVKKAL    IJl'LE    FOI{   TllK    PiUn.KSS.      First    CODlplUr    tlir 

vat  It  r  of  Fr  f)//  the  procesfi  rnr/itoifcd  in  ^  3Gf». 

Thru  rvpcdt  tJic  process,  nsin^  the  sitccessire  sums  oh- 
tai //('(/  iih  t/ic  first  process  instead  of  ihe  corresfmnd in 'J 
eoejjirirnts,  mid  stop])iiig  one  I'ni  hcforc  the  tost.  Tliv 
result  irill  In'  the  coeffwiiol  of  h. 

lirpeot  the  process  iriiji  the  neiv  sums,  stopjdn^  ijit 
one  trrnt  sooner.     Tiie  re.uf/i  irill  he  the  coeffrieut  of  Ir. 

Coiitiimr  the  ref)etition  frntll  n^e  have  the  first  tern} 
(full/  to  o])ernte  upon,  '  Ic/i  will  Itself  he  the  coejfieicnt 
of  the  hi  I 'h  est  ])ower  of  h. 


E: 


V    i.      r  le  example  above  given  is  performed  as  folloAv.-: 

+  3 


rocfRcientft 
Product  by  ? , 

First  sums, 
Second  j»roductB, 

Second  sums. 
Third  product, 


+  3  0 

J  14 

7  14 

i  ?! 

h  86 

4 

15 

2//3  4-ir)//  (-;w<  +  32. 


+  4 
28 

32 


Result,  Fi:l-\-h) 

Ex.  2.     In  the  I'nnction. 

let  ns  put  X  —  ;)  4   //,  and  exjnvss  tjic  result  in  powers  of//. 


s  than  that  of 

',t}(J  J'  —  r-\-fi, 

/. 

identiral  with 
of  tht'  forimr. 
tity,  whrrnis  ii 
riuLT  problem. 

expression 
iig  the  scpa- 


•ontputr  flir 

ii  rr  siu)it<  ob- 
)j'rrs})nn(liiiij 
r  last.      Tlir 

Sf0pj)7n[>    Ijrf 

cir.iU  of  Ir. 
he  first  term 
lie  cocffidcut 


cd  us  f()llo^?=: 

+  4 
28 

32 


a  EN  Eli  A  L    THEORY   OF  FQUATIONS. 


449 


('ootBcicnts,             3 
Products  by  3, 

-7 

+  5 
-3 

+  « 

+  6 

+  12 

-8 
+  54 

First  Hiiiiis, 
SeniiKl  products, 

-1 

+  0 

-f-2 
+  15 

+'4 
+  51 

+  18 
+ 105 

+  40 

S».'conii  .sums, 
Third  products, 

+  5 
6 

+  17 

+  55 
150 

183 

Third  sums, 

11 
6 

17 
6 

28 

50 

5! 

101 

205 

])(»\vers  of  //. 


Result,        F{^-\-h)  =  2/i'*  +  23/i'  +  10i;/"  +  205//'2  +  lH,']//  +  46. 

EXERCISES. 

1.  Compute  2//5 -f-2;j//« -f  l()l//''-f  :^0r)//2H-ls3/?  +  4f),  wlien 
//  =  X  —  3. 

2.  Compute  r^-^  _  r.t  -|-  7  for  .r  —  —  4  +  //,  —  3  -j-  //,  etc., 
to  +  3  +  h. 

Proof  of  the  Preredinrj  Process.  If  we  develop  the  ex- 
pression 

a  (Ji  +  /•)"  +  ^^  V'  +  '•)"  ^  +  r  (//  -f  r)«--  4-  f1  {h  +  r)"-^  -f-  etc., 

and  f'ollect  tlie  eoi'lVu-ients  of  like  j)owers  of  //,  wo  slutll  lind 

Coei".  of  //"      =  a, 

//«-»  =  nnr  -f  I), 

h"-^  =  {[^)ar'^-\-  {n  -  \)  fn- -i- r,  {A) 

//"-3  ^  /;|^„,3  +  {"~^)f>r^  4-  (/•-  -  ^)  cr  +  </, 

•  •  •  • 

•  •  •  • 
«-                               •                                                  •                                        • 

•  •  • 

fin-s  —  r'^jnr"  4-  (      ~    jbr''^  4-  (''-^  r,)'^'/'*"'  +  ete. 

Now  ex{iminin<r  Ex.  2  i);>eeding,  it  will  he  seen  that  we  ean 
make  the  eominitarion  by  nhimns.  iirst  eompuiing  the  whole 
left-hand  eohiniii  and  tins  ohtaininj;  the  eoetVieient  of //""', 
then  conipntiiig  tlie  next  eolnmti,  thus  obtaining  tbe  eoetli- 
cient  of  /i"~^,  and  so  on.  Commeneinp:  in  tbis  way,  and  n,*in^ 
the  literal  eoeiifieients,  a,  h,  c,  etc.,  and  the  literal  factor  r,  we 
t^hall  have  the  results 


,      W,      I,  , 


450 


GhJNEIiAI.    TIIKOHY   OF  EqUATlONS. 


a 


b 

at' 

c 

ar^  +    hr 

ar  +  b 

((,2  4-    hr  +  c 
2((r^  -f-    hr 

2m'  +  0 
ar 

dfif^  -\-  2hr  +  c 
3nri  +    hr 

Sar  +  b 


Gar-  +  '^hr  +  c 


I      J 


i<         t        • 


?^«y'  +  h 


m 


rr  «  is  the  (lc';i:rL'C'  of  tlio  cffUiitioii,  I  lion,  !»y  the  prccctliii^' 
process,  we  sliull  adtl  the  product  ar  to  h  a  times,  the  n  seiui- 


nile  sums  being 


ar-\-b,     2ar  +  h,     o((r-{-h,  ....  nar-\-h. 

To  form  the  second  cohniin,  we  multiply  each  of  these 
sums  except  the  last  l)y  r,  and  add  them  to  the  coelHeient  c. 
Tlio  terms  in  ar  ad(h'd  l)ein<jf  ar^,  2ar^,  3a/'~,  etc.,  the  >^\\m 
Mill  bo  (I -{- :i -(-:}  +  ...  .  -{-  /f  —  \)  ar'^.     The  coefiRcient  is  a  figii- 

rate  numbc  erpial  to  — i^'^p— -  (§§  ;»,S(;,  287).     The  sum  of 

the  coofTioionis  of  /;/•   is  n  —  1,  because   there   are  ?i  —  1  of 
them  used,  each  e([ual  to  unity.     Thereibre  the  tlnal  result  is 


{'.')^f>-'^+  Of'-  ^)^^r-{-c. 


which  we  have  found  (o  be  the  cooflicient  of  //'*~^. 

In  (his  second  column  the  partial  sums  or  coeilicienis  of 
ar''^  live 

1,    l-f:ir=3,   1-f 2  +  :}  =:G,  etc.,  to    1  f-3  +  3  +  .... +0^— -')• 

Therefore  the  numbei*s  successively  addeil  to  forir  the  co 
ellicicnts  of  ar^  in  the  thiixl  coluiuu  arc  1,  I  -j-li  —  t,   1  -f  IJ4  li 
=  U»,  cic.     The  coeflicients  of  bf^  will  be  the  suuu'.  as  those  of 
ar^  in  (he  column  next  piHCodin,i>\ 

(-'ontmuing  the  proivss,  we  see  that  the  coefTlclinls  lU'O 
formed  liy  puccessivo  addition,  as  in  jlni  following  table,  wheio 
each  number  is  the  sum  of  the  one  a'-  >ve  it  plus  the  one  on  ita 


<B^^:^^. 


WMNFMAL    TnKORY   OF   hVCATinXS. 


4rn 


^ 


1)  Ir  +  c. 

u'  nrccciliii;j; 
1,  the  n  aepa- 

■Ik 

nich  of  IIksc 

coellieicut  c. 

)tc.,   the-  sum 

lent,  is  a  iigu- 

The  sum  tit 

are  ??  —  1  <>f 
nal  result  is 


) 


roetheienls  ef 

ioYiv  the  eo 

-.  [,  I  flM-'"' 
me.  US  those  of 

mefTlclents  nro 

|g  table,  wiiero 

tlio  one  on  ha 


m 


etc. 
etc. 


etc. 


m     1 

¥       1 
¥       1 

^«     1 

etc.  etc. 


•4      10 


10 
:30 
etc. 


etc. 


'tc. 


•tc. 


etc. 


left.  We  have  earrird  tlie  table  as  far  us  ;/  =  ('»,  and  tlie  ex- 
pressions at  tlie  Ijottom  of  eacli  ei<hinni  will,  wlicii  ti  —  C,  he 
formed  from  the  numbers  in  this  tal)le,  taken  in  reverse  ordi-r, 
thus : 

Column  under  h,     iUfr  +     />»; 

il,  :>0^/r'+lO///''  +  4rr  +   il] 

p,  15r^r'+ lO^/'H'I'V'^.Vr  4-   c) 

/;     O^/r*  +    :)/»/•'  +  -Irr'  +  l^/z'-^  -f  'Zcr  +y' ; 


« 


it 


tt 


(( 


« 


(.' 


(t 


Now  the  numbers  of  the  above  selieme  are  tlie  tifjurate 
numbers  treated  in  §  287,  where  it  is  shown  that  the  n'^  num- 
ber in  the  i'^  column  after  the  column  of  units  is 


n{n  4-  l)Oi  +  2) 


{n  -f  /-I) 


l-'Z'l] 


=  (^i-')' 


Comparint^  witli  the  coeffii-ients  in  the  ecpuitions  (-/),  we 
-ee  that  the  two  are  identical,  which  i)roves  the  correctness  ot 
I  lie  method. 

370.  Application  of  the  Precfdinf/  Oprratioti  to  thr  Ex- 
tt'oction  of  the  llootx  of  Nnnicrical  Equatioufi,  Let  the  e([ua- 
tion  whose  root  is  k^  be  found  bo 


ax^  +  /Af»-i  +  cx^  '^  4- 


+  //  ^  <»• 


We  tind.  bv  trial  or  otherwise,  the  irreatciit  whoK-  number 
ill  the  root  x.     Let  ;•  be  this  number.     We  .substitute  r-f //  for 


452 


GEXKUAL    Til  1:0 It r   OF  EQUATIONS. 


\       i 


.r  ill   I  lie  above  expression,  and,  i)y  tin*  jjrcccdiiif,' ))rocc8s,  got 
an  ('i|uation  in  //,  nliieh  we  may  ])iit  in  tlie  form 

all"  +  ///i"  -»  -f  ^''//«  ^  +  r/'A"  -'*  -f  .  .  .  .  -I-  //■  r=:  0. 

Jit't  ;•' he  the  first  decimal  of  //,  We  put  r'-^li'  lor//  in 
this  ('(piation,  and,  hy  repeat irifj  the  proces>!,  get  an  equation 
to  determine  //',  which  will  he  less  than  0.1.  If  r"  he  the 
greatest  niiniher  of  hundredths  in  //',  we  put  //.'  =  r" -\-h',  and 
thus  get  an  ecjiiation  for  the  thousandths,  etc. 

371.  'I'Ih'  first  operation  is  to  liiid  the  number  and  appro\-- 
imate  values  of  the  real  roots.  There  are  several  ways  ofdoini; 
this,  among  whi(,'h  Slurni's  Thcorpin  is  the  most  celebrated, 
1)ut  all  are  so  lalxtrious  in  application  that  in  ordinary  eases  it 
will  be  found  easiest  to  i)roceed  by  trial,  substituting  all  entire 
numbers  for  T  in  the  ecpiation,  until  we  find  two  eonseeutive 
numbers  between  which  one  or  more  roots  must  lie,  and  in 
ditiicult  eases  ])l()tting  the  results  by  ^  345. 

It  is,  however,  necessary  to  be  able  to  set  some  limits  h  •- 
tween  which  the  roots  must  be  found,  and  this  may  be  done 
by  the  following  rules: 

T.  .  ///  rtiiiatioii  in  which  ull  thr  cnrfjiciciits,  indudiirj 
(he  iihsohttc  term,  are  positire,  caui  have  no  positive  nuil 
root. 

For  no  sum  of  positive  quantities  ean  be  zero. 

II.  //'  ///  comjnUing  the  vdlue  of  Fx  for  any  assnninl 
])ositiir  value  of  x,  hy  the  process  of  §  3fi6,  ive  find  all  the 
sums  j)ositirr,  there  can  he  no  root  so  great  as  thut 
assumed. 

For  the  substitution  of  any  greater  num])cr  will  make  all 
the  sums  still  greater,  and  so  will  carry  the  last  sum,  or  /v, 
still  further  from  zero. 

III.  Tf  the  sicDis  are  alternately  positive  and  negn- 
tive,  the  value  of  x  ive  employ  is  less  than  any  root. 

IV.  //  two  values  of  r  6ii^e  different  signs  to  Fx,  there 
must  he  one  or  some  odd  number  of  roots  between  these 
values  (comjiaJN'  ^  ^U.l). 


GENEliAL    rUEOnr   OF   hVl'ATlOXfi. 


4C)'3 


V.  Th'o  raJurs  of  x  wliirli  hunl  to  fJir  soiiir  siri/t  of  /•> 
include  cither  no  roots  or  an,  cren  nitntfjrr  of  mots  f}c- 
tivccn  them. 

Lot  us  take  as  a  first  cxaiiiijlc  tlic  ("filiation 

2-3  -  T^  +  7  =  0. 

Let  us  first  assume  or  =  4.     We  compute  as  follows  : 

Cooflk'ieiits,  1  0  —7  +7 

Products,  _4  16  36 

Sums,  +4  +!>         -f-43 

So  F  (A)  —  +43,  and  as  all  the  cooHicicnts  arc  positive, 
there  can  he  no  root  as  great  as  4. 

Ptiltin*,'  Xz=—\,  tlu!  sums,  iiiclu(iiii«i:  (he  first  coeflicient. 
1,  are  I,  —4,  +9,  —V.).  Tlu-se  hein;;  iiltcrniitcly  positive  and 
nep^ative,  there  is  no  root  so  small  as  —  I. 

Suhstitiitiii<;  all  inte^xers  between  —4  and  -(- 4,  we  lind 


/'(-4)  = 

/'(-;})  =4-1, 

/'(-:>)  =    +i:5, 

F{-\)  ^  +1:5, 


/•'(I)  -  +  I, 

F{'i)  -  +    1, 
/'(:))  =.  +1:3. 


w 


If  we  draw  the  curve  correspond ini;-  lo  these  values  (Jj  ;J4r)), 
e  shall  find  one  root  hetween  —'.\  and  —4,  and  verv  near 
—3.0.'),  and  the  curve  will  dip  helow  the  base  line  hetween  -f-  I 
and  +2,  showing  that  there  are  two  roots  ht'tween  these  niiin- 
l)ers  ;  that  is,  there  are  two  roots  of  the  form  1  +//,  h  heinir  a 
positive  fraction.  Transforming  the  equation  to  one  in  //, 
by  putting  \  -\-  li  for  x,  we  find  the  equation  in  //  to  be 

7^3  ^  3/^2  _  47/   f  1  :^  0.  (1) 

Substituting  h  =  0.2,  0.4,  O.C.  O.S,  we  find  that  there  is 
one  root  between  0.3  and  0.4,  and  one  between  0.0  and  0.?. 
Let  us  begin  with  the  latter. 

If  in  the  last  equation  we  put  /t  —  ()J\-irh\  we  Mnd  the 
transformed  equation  in  //'  to  be 

Fh'  —  h'^  -\-  4.8/; "-J  +  O.r.8//  -  0.104  =  0.  (2) 

If  we  substitute  different  values  of  //'  in  this  equation,  we 
^9 


454 


OKNEUAL    Tl/hVItr   OF   FJjUATIONS. 


S 


bIi;iII  fitiil  that  it  mu«t  cxcct'd  .<)'.»,  and  uh  it  must  l)o  Icsh  tliuti 
0.1,  wo  conclude  that  It  is  the  li«(ur('  souglit,  and  put 

W  =  XV,)  4-  //■'. 

Transforming  tlie  c([uation  (:,*),  wo  lind  liic  eciuation  iu  /t' 
to  he 

h"^  -f  .J.OTA' '^  -f  i.o(;8;j/< '  —  o.oo;u'ji  -  o.         (:5) 

Since  h"  is  necessarily  less  than  0.01,  its  lirnt  iligit,  whieh 
is  all  we  want,  is  easily  found,  hecause  the  two  tirgt  tenns  of 
the  ei|iiation  are  very  snuill  compared  with  the  third.  So  we 
Biniply  divide  .OO.'jr.H  hy  l.nfjs:],  and  tind  that  .00::i  is  the  re- 
quireil  digit  of  h".     We  now  put 

h'  =  .002  +  //'", 
and  transform  again.     The  resulting  e(|uation  for  A"  is 

h""^  4-  ."i .<»:(;/< "-J  -1^  i.5.sH5i)2A"'  —  o.oo()o;ui  12  =  o.    ( i) 

The  digits  of  ./•,  //,  //',  and  h"  which  we  have  found  show 
the  true  value  of  .c  to  he 

X  =  1.002  +  //". 

Hy  continuing  this  process,  as  many  tigures  as  we  please 
may  he  found,  liut,  after  a  certain  i»oint,  the  operation  iH.iy 
be  ab])reviuted  hy  cutting  off  the  last  tigures  in  the  coetllcients 
of  the  powers  of /^ 

Tlic  work,  so  far  as  we  have  performed  it,  may  be  arranged 
in  the  following  form  (see  next  page). 

The  numbers  under  the  doidtle  lines  are  the  coefllcients  of 
the  powers  of  //,  h\  h",  etc.  It  will  he  seen  that  for  each  digit 
we  add  to  the  root,  we  add  one  digit  to  the  coetllcient  of  //-, 
two  io  that  of  //,  and  thi'ce  to  tlie  altsolute  term.  We  lunc 
thus  extended  the  latter  to  nine  places  of  decimals,  which,  in 
most  cases,  will  give  nine  figures  of  the  root  correctly.  If  litis 
is  all  we  need,  we  add  no  more  decimals,  hut  cut  oflf  one  fioin 
the  coetllcient  of  //,  two  from  that  of  A^,  and  soon  for  cad) 
d'^cimal  we  add  to  the  root. 

We  shall  tind  the  next  figure  after  1.092  to  be  zero  ;  so  wo 
cut  off  the  figures  without  making  any  change  in  the  eoclH- 
cients.  The  next  following  is  2,  so  we  cut  off  again  for  if,  and 
multiply  as  shown  in  the  following  continuation  of  the  process: 


i 


(fKy/'JUAL    TllHOIiY   (tF   h\fl'.lTI(»yS. 


bo  less  thttti 

)Ut 

luatiou  iu  /t'' 


:  0. 


CO 


lirst  lenns  ol' 
liinl.  So  we 
0*^  is  the  rc- 


r  h'"  is 

VZ  :=  U.     (I) 

c  found  show 


us  wf  pleiise 
)|K'nitioii  niJiy 
lie  coL'llicii'iits 

ly  be  iin'iin<re«l 

cocfllcients  (tf 
lor  eiicli  dii^it 
L'llieieiit  of  //-, 
riTi.  We  bii\(' 
nils,  whicli.  ill 
vctly.  ll'iliis 
(ifV  our  I'lKiii 
s(»  on  lor  each 

)0  zoro  ;  so  wo 
ill  the  coctli- 
siiii  i'ur  it,  and 
o['  the  process: 


0 

-7 

+7 1  \.im 

+  1 

+  1 

-i\ 

Tl 

-tt 

+  1  (KIO 

+  1 

+3 

-1.104 

+  2 

— 4.(M) 

-   .104000 

+  1 

+  2.HJ 

+    .KMIHOO 

+  3.0 

-I.H4 

-  ,n!5;{i<.M(KK) 

-f  .6 

+  3.53 

+  .(M):{ir,«{MMS 

+  :!:« 

-f-0.«lH(M> 

-^4112 

.0 

+  0.4101 

4.8 

+  l.l30i 

6 

+   .4-IH3 

■+4.«0 

Tr.r)«iH;{(M) 

» 

l(H44 

~~4.8ir 

+T.."7stll 

0 

101  IM 

4?i»H 

'.m.     ' 

__     9 

+  1.5885U3 

■+.'>.  070 

"7)072 

2 

5.074 

+  5.07G 


CONTINUATION  OP  moCESS. 


+ 15.070 


+  1.5885;9  3 
I 

1.5887 
1 

-;{4ii2 
:U774 

15MI) 

I.5|8,8|8 

-  740 

"— li:{ 
111 

—^i 

031471 


It  will  be  seen  that  fmm  this  point  we  make  no  use  of  the 
(•oinieient  I  of//-',  aJid  only  with  the  seeond  (hciuial  do  we  use 
I  lie  eoellieieiit  of  /r.  After  that,  the  ivinainiii^i,'  four  iiirures 
:u'e  obtained  by  pure  division. 

There  is  one  thintr,  however,  which  a  computer  should 
always  attend  to  in  multiplying  a  number  from  which  iic  has 
cut  oflf  tigures  in  this  way,  namely: 

.llirai/fi  cnrvii  to  fhr  jn-odurt  thr  nniiihrr  ic/iirJi  would 
hare  hccii  carried  if  the  jiilures  had  iiol  Itren  cut  off,  and 


Vi 


^ 


dl 


A 


-c^i 


s^ 


/ 


^^ 


^ 


/^ 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


1.0 


2.8 


I.I 


S  IM 

140 


1.25 


12.5 

IIM 

IIIW 
U    IIIIII.6 


Photographic 

Sciences 
Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  N.Y.  14580 

(716)  872-4503 


A 


V 


<^     ^4£     M  ^ 


^g. 


f     - 


^o 


^ 


V 


\ 


\ 


^\^ 


6"^ 


«^. 


^^\^4> 


456 


GENERAL    THEORY   OF  EQUATIONS. 


ft 


I 


increase  it  liij  1  //  the  figure  following  the  one  carried 
would  have  hecii  5  or  greater. 

For  instance,  we  hi'd  to  ninltiply  by  7  the  number  15i888. 
If  we  entirely  omit  the  figures  cut  off,  the  result  would  be  lOo, 
But  the  correct  result  is  lll|;il6;  we  therefore  take  111  in- 
stead of  105. 

Again,  in  the  operation  preceding,  we  had  to  muldply 
158i88  by  4.  Tlie  true  product  is  G35,52.  But,  instead  ol' 
using  the  figures  635,  we  use  636,  because  the  former  is  too 
small  by  |5^,  and  the  latter  too  great  by  |48,  and  therefore  the 
nearer  the  truth.  For  the  same  reason,  in  multiplying  1.588,8 
bv  1,  we  called  the  result  1589. 

Joining  all  the  figures  computed,  we  find  the  root  sought 
to  be  1.69;>02U71. 

Let  us  now  find  the  negative  root,  which  we  have  found  to 
lie  between  — 3  and  — 4.  Owing  to  the  inconvenience  of 
using  negative  digits,  and  thus  having  to  change  the  sign  of 
every  number  we  multiply,  we  transform  the  equation  into  one 
having  an  equal  positive  root  by  changing  the  signs  of  the 
alternate  terms.    The  equation  then  \s  x^  —  Ix  —  7  =  0. 

The  work,  so  far  as  it  is  necessary  to  carry  it,  is  noAV  ar- 
ranged as  follows : 


0 
3 

8 
8 

6 
3 


-7 
18 


-7 
6 


3.0489173395 


9.00 
_4 

9.04 
4 

9.08 

_jL 

9J[20 

8 

9.128 

8 

9.136 

8_ 

19.1144 


20.onoo 

.3616 

20.3616 
.3632 

20.724800 
73024 

20.797824 
73088 

20.8709112 
82310 

20.87914|2 
823 

2078873^ 
9 

201.8181715 


-1.000000 
814464 

-0.185036000 
.166882592 

-     .19T53408 
18791228 

-362180 

208875 

^53305 
146213 

^'092 
_6266 

-8'26 
627 

-199 

188 


■11 


NS. 

0  one  carried 

limber  15:888. 
would  be  105. 
e  take  111  iii- 

1  to  miUiiply 
;iit,  instead  ol' 
3  former  is  too 
L  therefore  the 
iplying  1.588i8 

he  root  sought 

have  found  to 
L'onvenience  of 
ige  the  sign  of 
nation  into  one 
[le  signs  of  the 

—  7  =  0. 
y  it,  is  now  ar- 

.0489173395 


000 
404  ___ 

580000 

882592 

153408 
1791228 

362180 

2(»88T5 

T53305 
146213 


-7092 
J266 

-826" 
_637 

'-199 

188 

-11 


G  EX  Eli  A  L    THEORY  OF  EQUATIONS. 

The  negative  root  of  the  equation  is  therefore 

—  3.048'Jir;j;)!)5. 

EXERCISES, 

Find  the  roots  of  the  following  eqnations: 
I.     .9-3  _  3.^2  -(-  1  =  0  (;3  real  roots). 

2. 

3- 

4- 

5- 


45 


r.f 


x^  —  3x  +  1  iiz  0  (;}  real  roots). 
.7-4  _  ix^  -\-  2  —  0  {•}  positive  roots). 
.r2  -f-  .1;  _  1  —  0. 


Prove  that  when  we  change  the  algebraic  signs  of  the 
alternate  coeflticients  of  an  equation,  the  sign  of  the  root  will 
be  changed. 

373.  The  preceding  method  may  bo  applied  without 
change  to  the  solution  of  nunu>rical  (juadratic  equations,  and 
to  the  extraction  of  square  and  cu])e  roots.  In  fact,  the  S(juare 
root  of  a  number  w  is  a  root  of  the  ccjuation  x-  —  n  =  0,  or 
x'^  -{-Ox  —  n  =  0,  and  the  cube  root  is  a  root  of  the  equation 
x^  +  02-2  -\-Qx  —  n  =  0. 


Ex.  I.   To  compute  V^. 


0 
1^ 

1 

1_ 

27o 

0.4 

2.4 
_4_ 

2":80 

1 

2.81 
1 

27820 
4 

2.824 

2^8280 
2 

2.8282 


-2 
1 


1.41421356 


-1.00 
_  .90  _ 

-70400 

___281_ 

-11900 
11296 

-60400 
56564 


-3836 

2828 

-1008 
__8J9 

-159 
141 

-18 
17 


H  ■ 


fi    <   ;  n  A 


•■j 


S|.8!3  8i4 


4S8 


GENERAL    TUEORT  OF  EQUATIONS. 


? 

i! 

} 

Ex.  2.  To  compute  the  cube  root  of  0842036. 


0 

0 

-9843036  1  314.30303343 

2 

4 

8 

2 

4 

-1843 

2 

8 

13(51 

4 

1300 

-"581036 

2 

61 

539344 

60 
1 

1301 
63 

4il'M3000 
41374307 

61 

-417793 

133300 

413338 

1 

63 

3536 
134836 

4467 
4133 

1 

3553 

334 

630 

137388.00 

376 

4 

103.09 

58 

634 

T37580.69 

55 

4 

193.78 

3 

638 
4 

137773.47 

1.93 

64370 
.3 

131717|75|4 

643.3 

3 

643.6 

3 

'643.9 


'.V, 


I)  ' 


214.30303243 


55 
3 


APPENDIX. 


SUPPLEMENTARY   EXERCISES. 


lii 


r 


Note.  Tlie  following  additional  exercises  and  problems  are  of  the 
same  general  character  with  those  in  the  body  of  the  book.  Tliey  are 
partly  original,  and  partly  selected  from  'he  best  recent  German  col- 
lections of  problems.  They  are  arranged  under  the  section  numbers 
to  which  they  pertain,  so  that  the  teacher,  on  arriving  at  those  sections, 
will  be  able  to  select  as  many  of  them  as  he  deems  necessary  for  the 
drill  of  his  class. 


f 


SUPPLEMENTARY  EXERCISES. 


Algebraic  Atltlitiou  aiul  Subtraction. 


§15. 

Supposing  one  to  start  from  a  certain  point  on  tlie  scale 
of  numbers,  and  then  move  over  positive  and  negative  spaces 
as  follows,  It  is  required  tc  find  his  stopping-point  in  each 
01  the  loilowing  cases: 

1.  Starts  from  +  4,  and  moves  through  +  2  —  3-|-9--7 

—  2  units. 

2.  Starts  from  +  9,  and  moves  through  -  l  -  g  -  9  4-  5 
+  8  units. 

3.  Starts  from  —  1,  and  moves  through  +  2  —  3  +  4-5 
+  6  units. 

4.  Starts  from  —  8,  and  moves  through  —  l  +  3_5_i_7 

—  9  units. 

5.  Starts  from  —  12,  and  moves  through  —9-6  +  8  +  5 
+  8  units. 

§31. 

I.  How  far  is  A  from  B  (positively  or  negatively)  when 
they  have  severally  made  the  following  motions  from  the  same 
point  on  the  scale  of  numbers: 

A ,  , B , 

a.  -2-3-5  +  7.  +1  +  2  +  3  +  4  +  5. 

b.  -5  +  5-6  +  G.  +5  +  6-2-4  +  12. 

c.  -2  +  7  +  8  +  9  +  10.  -7-3  +  4-5-6. 

./.   -1-2  +  6-2-1.  +3  +  4  +  5-8-3.    Ans.~l 


*1 


In 


4(52 


ALUEBliAlC  ADDITION 


\ 


2.  What  is  the  int'iiuing  of  the  following  expressions  : 

That  man  is  —  6  years  older  than  his  wife  ? 
Kichmonii  is  —  70  miles  uorth  of  Washington  ? 
You  arc  —  3  inches  taller  than  your  brother  ? 

3.  The  Autocrat  of  the  Breakfast  'J  able  tells  of  a  Par^oi' 
Turrel  who,  dying  in  the  last  century,  bequeathed  a  noted 
chair  to  the  oldest  member  of  the  Senior  class  in  llarvanl 
College,  which  was  to  be  passed  down  from  class  to  cIjihs 
indelinitely.  Tiie  first  Senior  who  got  it  was  to  pay  5  crowns, 
but  each  succeeding  one  was  to  get  it  at  a  price  1  crown  ks.s 
than  that  paid  by  his  predecessor.  How  would  the  require- 
ment of  the  will  work  at  the  end  of  7  and  of  100  years? 


§34. 

I.  Find  the  value  oi  a  —  b  and  of  ^  —  rt  when  a  and  h  have 
the  following  sets  of  values  : 

(1)  (2)  (3)  (4)  (5)  (G)  (7)  (8)  (9)  (10) 
«=  +2,  +7,  -0,  -5,  -17,  +  8,  -33,  -18,  +12,  +22 
h=   -3,  -9,  -3,  +8,  -29,  +14,  +13,  -19,  -12,  -22 


a-h=   +5 
h—a=   —5 


2.  Com])ute  the  values  of  1  +  3a:  and  of  1  —  3^  for  the 
following  11  values  of  x  : 

a:  =  -  5,  -  4,  -  3,  -  2,  -  1,  0,  +  1,  +  2,  +  3,  +  4,  +  5. 

3.  Compute  the  values  of  «  +  25  and  of  a  —  26  for  cacli 
of  the  10  sets  of  values  of  a  and  b  in  Ex.  1. 


§56. 

1.  How  much  is  a  +  2a;  greater  than   a  —  dx,  and  vice 
versa '? 

2.  How  much  \s,  a  —  b  greater  than  b  —  a? 

3.  TTow  much  is  0  greater  tluin  a  —  2b  ? 

4.  How  much  is  0  greater  than  —  x?    Than  +  a;  ? 


AM)  SUUTliACTlUN. 


403 


ressions : 

nfu  ? 
liiif^ton  ? 
•other  ? 

Is  of  :i  Pai'M)!' 
ithed  a  noted 
5S  in  Jliu'vaitl 
class  to  cliiss 
I  pay  5  crowns, 
je  1  crown  Ics.s 
;1  the  rcquirc- 
lO  years? 


n  a  and  h  have 

3)      (0)     (10) 
18,  +12,  4--2-i 


0, 

—  v>   - 

•    ■ 

•    •    •    •         • 

-  dx  for 

the 

3, 

+  4,+ 

5. 

■  26  for 

eacli 

-  Zx,  and  vice 


-n  +  a;  ? 


5.  A  party  of  9  boys  were  formed  into  a  solid  square  of 
^  rows,  with  3  boys  in  each  row.     The  rear  Icfi- 

liaiid    boy   15   was  t  inclies  tall.     Kvery  othei    boy 

was  ./•  inches  taller  tlian  the  boy  next  behind  hinj,     ,,  '     *     * 

and  y  inches  shorter  tlian  the  boy  on  Ins  left.    Kx- 

})ress  the  lieight  of  each  boy,  and  the  sum  of  the  heights  of 

all  the  boys. 

6.  During  six  successive  days  a  man  earned  m  cents  more 
every  day  than  he  did  the  day  before,  and  i)aid  out  n  cents 
less.  On  the  first  day  his  earnings  were  h  cents,  and  his  pay- 
ments k  cents.  IIow  much  had  he  left  at  the  end  of  the 
sixth  day? 

7.  Of  two  travellers,  X  went  east  k  miles  and  then  returned 
h  miles  toward  the  west ;  Y  went  west  x  miles  and  then 
returned  y  miles  toward  the  east.  If  they  started  together, 
how  far  was  X  east  of  Y  when  they  stopped  ?  llow  far  was 
Y  east  of  X  ? 

8.  There  were  three  travellers  on  the  same  road,  15  being 
X  miles  west  of  C,  and  C  y  miles  west  of  A.     A  went  m  miles 
toward  the  east ;  B  went  twice  as  far  as  that  toward,  the  east; 
and  C  went  4m  miles  toward  the  west.     IIow  far  was  ear' 
west  of  the  two  others  when  they  sto])ped  ? 

9.  Of  two  men,  A  and  B,  A  had  a  doUars  and  B  had  x 
dollars  on  Monday  morning.  On  Monday  evening  A  paid  B 
d  dollars,  and  B  returned  y  dollars  of  tiiis  to  A.  Eacli  fol- 
lowing evening  during  the  week  A  paid  B  g  dolbrs  less 
than  before,  and  B  returned  A  z  dollars  less  than  ho  did  tlie 
evening  before.  IIow  much  had  each  on  each  morr.ing  from 
Tuesday  to  Saturday? 

10.  Four  casks,  marked  A,  B,  C  and  D,  each  containing  r 
gallons  of  water,  stood  at  the  corners  of  a  square.  I'hon  m 
gallons  were  poured  out  of  A  into  B,  n  gallons  out  of  B  into 
C,  p  gallons  out  of  C  into  D,  and  q  gallons  out  of  I)  into  A. 
IIow  much  was  then  in  each  cask  ?  Prove  the  result  by 
showing  that  the  sum  of  the  quantities  in  all  the  casks  is  4r. 

II.  The  same  four  casks  at  first  contained  a,  h,  c  and  d 
gallons  respectively.  Then  x  gallons  were  poured  out  of  B 
into  A.     Then  a  quantity  equal  to  what  was  left  in  B  was 


4G4 


A  LUhliUA  /(J  AJJJjJnuX 


poured  from  0  into  15 ;  a  (luantity  fcjiuil  to  what  was  left  in  {] 
WHS  pourcMl  fntiii  1)  into  C  ;  ami,  liually,  Ji  (iiuiiitity  equal  to 
wluit  was  k'fL  ill  I)  was  poured  from  A  into  J).  J  low  mueli 
was  then  left  in  each  cask  ?     I'rove  as  before. 

12.  Tliree  traders,  A,  ii  and  C,  had  a,  b  and  c  dollars 
res{)ectively.  A  bought  c  dollars'  worth  of  goods  from  \\  ;  li, 
a  dollars'  worth  from  C  ;  and  C  bouglit  b  dollars' worth  from 
A.  When  each  ]»aid  the  other  for  the  goods,  how  mueh 
money  had  each  left  ?  What  was  the  sum-total  of  money 
j^ossessed  by  the  tliree  ? 

13.  Giveu  a  quadrangle  the  lengths  of  whose  sides  arc  a, 

b,  c  and  d  respectively.     Enough 
of  the  side  b  is  cut  o(f  and  added 

'  to  a  to  double  the  latter;  (he  re- 
mainder of  /;  is  then  doubled  bv 
cutting  of!  from  c ;    and  the  re- 
mainder of  c  is  doubled  by  cnttiuf 
off  from  d.     How  long  will  each  side  then  be  ? 

14.  Of  two  men  starting  out  from  the  same  point,  A 
walked  m  miles  west  the  first  day,  and  k  miles  more  each  fol- 
lowing day  than  he  did  the  day  before  ;  B  walked  p  miles 
west  the  first  day,  and  x  miles  less  each  day  than  he  did  the 
day  before.  How  far  was  A  west  of  B,  and  how  far  was  B 
west  of  A,  at  the  end  of  the  first,  second,  third  and  fourth 
days  respectively  ? 

15.  If,  on  this  line,  we  suppose  the  point  B  to  be  at  the; 

B  C 


East. 


West. 


distance  h  west  of  A,  and  C  to  be  ut  the  distance  c  west  of  A, 
then,  in  .'ilgebraic  language: 

IIow  f:ir  is  A  west  of  B  ? 

How  far  is  A  west  of  C  ? 

How  fur  is  C  west  of  B  ? 

How  for  is  B  west  of  C  ? 

How  far  is  the  middle  point  between  B  and  C  west:  of  A  ? 

How  far  is  the  middle  point  between  C  and  A  we?t  of  B  ? 


CLEAiiiya  or  iwiiKyriirsh's. 


105 


B  to  be  at  t!u! 


IIow  far  is  tliu  middli!  point  hi'tweeii  A  ami  \\  west  of  ('  ? 
What,  is  the  algobniiu  suiri  of  tlit'su  last  tlircc;  (lislincts  'f 

NoTR.  Should  tlu!  stiidciil  fmd  luiy  dilliculty  in  this  or  the  iir\t 
(iwt'slioii,  he  slioiild  bci^iii  by  cxiircsr-iiiu:  the  disliiiiccs  a  and /mii  nuiii- 
bur.s,  mid  iioUciuLj  the  processes  by  wliich  {\\v  nuasu'vs  are  found. 

1 6.  'iMio  three  points  A,  B  and  C  arc  at  the  respective 


i 


B 


"  i 


I  I 

distances  n,  h  and  c  west  of  a  fourth  jioint,  M.    Express  alge- 
braically the  three  distances 

B  west  of  A  ;  A  west  of  C  ;  C  west  of  B, 

and  take  their  sum.     Ex[)ress  also  the  distances 

A  west  of  the  middle  point  between  M  and  B, 


B 
C 


ii 


a 


(( 


i( 


tt 


M  and  C, 
M  and  A, 


and  find  the  sum  of  these  three  distances.     Then  express 
A  west  of  the  middle  point  between  B  and  C, 


B 
C 


a 


(( 


(( 


(( 


tt 

tt 


0  and  A, 
A  and  ]i. 


and  find  the  algebraic  sum  of  the  three  distances.      Express 
also  the  three  mutual  distances  between  the  middle  points  of 
lines  AB,  BC  and  CA  respectively — that  is  : 
Mid.  point  betw.  A  and  B  west  of  mid.  point  betw.  B  and  C, 


etc. 


etc. 


etc. 


ce  c  west  of  A, 


§61. 

Clear  the  following  expressions  of  parentlieses,  and  com- 
jino  the  terms  by  addition: 

1.  3  7)1  —  [h  —  2m  —  (h  -\-  rn)  —  {h  —  m)]. 

2.  {a  —  d)  —  {a  -\-  b)  -\-  {a  —  7n)  —  {a  —  ni). 

3.  ^a  +  b)-{a-  h)  -  [{a  -  b)  -{a  +  /.)]. 

4.  2h  -  {3/i  -  \Jch  —  (5/i  -m)-\-  m]  +  2m\. 

5.  3c  -  2(1  -  {2(1  -  Sc)  +  [-  {c  -  (I)  -  (3r+  2^Z)]. 

6.  4A  -  7/i  -  (4A  +  7w)  -  [3/i  +  {-im  -  (i)  -  (oz/i  +  h)]. 


1 1. 


m 


4C0 


MUl/nriJCATION. 


h.l 


If 


I-    '* 


7.  x+\x-  a-  {}ln  -  'ix)  -\-  \(t  —  {a  -x)]]. 

8.  Gfl  +  I  ha  -'■Zx-\-\  \a  - '6x  -  (3«  -  4a:)  j  | . 

9.  {a  f  h  -  <;)  ^{n-b-{-  c)-\-{-a  -\- h -^  r)-{ft  -f-  /y  -f  r). 
« o.  nr  -f  dx  -  i'^a  +  :>./■)  -  {;ia  -f  a,')  -  I  ^^  -  {a  -  ./■)  |. 

II.  /;  +  [  -  ii^  -  :i6'  -  (3r  -  Z-)  -  ;j  /y  j. 

12.  -[- (;}m-^>/<)-h('i//<-;i//)j  +  [{r>m-4n)-{'.]N-\iN)\. 

Muitipilciitioii   iuul  A<lcntio]i. 

§74. 
Clear  the  following  expressions  of  parentheses: 

1 .  /j\a  {c  —  x)  +  b (.-; -\- x) -\-  ax  [d  -  c{x  -a)]\. 

2.  »f  \x  -  n  (/>  -  y)  -^b{n-\-  y)  +  y  {n  +  b)\ . 

3.  <ni  [an  (1  —  a^;)  -f-  (I'n'*  (1  —  «7i)]. 

4.  h\i^h[i^-h{l-{-h)]\. 

5.  a;  |1  —  a;  [1  —  a;  (1  —  x-)](. 

6.  X  \p -^  x\fi-\-x{r -{-x)']]. 

7.  a;{;;-a:[//  -a:(r-a:)]i. 

8.  n\[{ax-{-b)x-\-c]x-\-(l\. 

9.  <<:  { [(rt.r  —  b)x  —  6']  a:  —  (/|. 

10.  p  \[{px  -^  p')  x  -^  p']x  -\-  p'\, 

11.  I  [(//?a:  —  w')  a;  —  ?;i']  x  —  7>i*  |  war. 

12.     [ce  {b  -  C)  +  b'  {C  -  «)  +C^  (rt  -  Z.)]  ^7Z>6'. 

13.  m  [a'  (a;  +  y)  -^  b' {x  -  y)  ~  x{a'  +  b')]  +  (a'  -  b')  ym. 

14.  a  1^^  —  Z*  [rt  —  c(«  —  (/)]}. 

i6.   («  +  a-)(^-y/)  +  (a-a:)(i  +  2/). 
1 7.   (?«  4-  n)  (a;  +  ^)  —  (m  -  n)  {x  —  y). 

(§  '76.) 

Arrange  the  following  expressions  according  to  powers 
of  a: : 

1.  (,;'_a;»4-l)a'4-(a:'-a;H-l)rt'+a\ 

2.  l  +  a;-a;"-a;'-«(l+a;-a;»)  +  r7'(l  -  a-)  -  a'. 

3.  ma'~na\x  -  1) -?/i«'(a;'-  a;  +  1) -  wa(a;'-a-'+a:  ~  1). 

4.  a  —  X  [b  —  x\c  —  x  [d  —  x)'\\. 


f  (a'  -  b')  ym. 


ing  to  powers 


DIVLHION. 


4(57 


Write  out  the  rosiil'^s  nf  n.r.  /•  n 
bluets  on  sight:  ^'""^'"'^  i^«^^^'«  ^"^1  i'l'- 


'.   {ax-^b,/)\ 
7.  (^^'^ +  ;%)». 

II.  x{x-^y)\ 

<3.   «(^-;y)(:r  +  y) 


2-     {flX-hn)\ 

6.   (f(,r  -  i>b,^)\ 
14.   '^'M'^  -  ^)  («  +  .r). 


17.  (a  +  by-\-(a-~b) 


3w/  +  u)  (ihn  ~  n). 


Form  the 


VJll 


r8.   («  + /^)»  _  („  __ /^^ 


according  to  powers  of 


^Inesof  the  following  (quantities. 


^,  y  and  5; 


and 


iir'JUinre 


20.    {ax  +  dy)«  _  (/,,.  _  ^yy 


(^^^'^  -  nyf  -  (,/,:^  __ 


//)' 


24.   {ax  -^byJ^  cz)  {ax  +  by  -  ,,) 


(.^r  -  /;^  -|.  ^^)   (,,^.  _  ^^  _  ^_^^^ 


I.  6«V^c' -4- 2«a:'^. 


Wvision. 

§(S5.) 


24a 


x'Uj  ~  {:>a'x 


3,1^2 


y 


rn 


3.  12««.c='y  -^  4f;V_?/ 


4.  ff" 


T{c-^d)-^ax{c-{-d). 


S.  rr(:c-y)-^,^(^_^^^^ 


468 


FACTORING. 


:Wi 


l'  *' 

.1 


6.  ^6-  (a;  +  y)  -T-  6»  (:?;  +  y). 

7.  a;"?/  (rt  —  ^)  -^  X  {a  —  b). 

8.  10.^^  (a  -  h)  +  G.?;'  («  +  ^')  -^  ^. 

9.  .T"  +  ^  -  .r"-^  +  2-"-2  +  2;"  -T-  a;"-'. 

10.  10  (a  +  by  - 15  (rt  +  by  - 10  (a  +  Z*) 

11.  0(a  -\-by  ^4.{a-\-by. 

12.  {a  +  .i-y  {a  +  yy  -4-  («  +  x)*  {a  -\-  yy. 

13.  -  12rt"'^" -^  4rf  ^^'^. 


5  (rt  +  J). 


14.  « 


m  +  l 


a 


m  +2 


a 


n»  +  3_^^3 


Factoring. 

§89. 

Factor  the  following  expressions: 

I.  ab'c'  +  rt^Z^'c  +  a'bc'  -  a'b''e\ 


2. 

.T  —  a.x!^  +  J.c\ 

3- 

—  ??i'  +  Wi';i  —  w^/i' 

4- 

c?;//-  —  ci^m'^  +  ^f'm'. 

5- 

C"/;""  —  c-'^Z*". 

6. 

—  abc  -{-  fii'abc. 

7- 

a'a;"-«V-'. 

8. 

ab'x'y'  -  a*b\v'y. 

9- 

7WW'p'  —  Sw'w'jO. 

10. 

2a"x-"''  -  7<i"*.r". 

II. 

8«i6'  -  12a'Z>V'. 

§91. 

In  the  following  exercises,  first  take  out  all  monomiiil 
factors  common  to  the  several  terms,  as  in  §  89,  and  factor 
the  remaining  terms  by  the  rules: 


I. 

a'  -  ab\ 

3- 

im'^x  —  dii'x. 

5- 

a\€'  -a\ 

7. 

iifx^  —  m^x. 

9- 

a'x'  -  ■^a'x'. 

II. 

IGmy  -  25^^?/' 

13.  ain^  —  a^  -f-  Ha'b  —  ab^. 
15.  ax""—  4«'.?-"  +  4rt'. 
17.  iyx'  -  nx*y'-\-iKr.y. 


2. 

W'w"  —  71*. 

4. 

mx^  —  in. 

6. 

ax^  —  ax. 

8. 

9.r'  -  4x*. 

10. 

?w'y'  —  wt'y'. 

12. 

49ff'x'  -  ma*x. 

14. 

a'x'  -  4b'x'  +  ibex'  -  ^-'^z'. 

16. 

a'Z>  -  4aV)'  +  4r/^*^ 

18. 

4.c^7/''  +  V2x'y'  4-  9.r^7/\ 

93. 


1.  2  (.T*  +  ?/*  +  z')  -  4 (.x-y  -I-  y'z'  +  ^V). 

2.  a*  +  lOZ**  +  6'^  -  Sn'b'  -  8b'c'  -  ^c'a'. 


f  i)  -  5  («  +  h). 


e  out  all   monomiiil 
in  §  89,  and  factor 


X. 

x\ 

taV;^  +  4f/Z*\ 


^'  +  z'x'). 


PACTOn/XG,  . 

3.  2  (rr'y'  +  f,^  +  ^r^.)  _  ,^,  _    ,  _  ^, 

4.  8a^b^  +  32^V-'  +  8r«^  -  a^  _  le^^  J  igc\ 

§94. 


J.  a'-i-  4«V  +  4«^;r\     Ans.  a^  (a  -f  o ,n. 

A?is.  a   {a  +  x)  (a  -  .r)  (,,'  _|_  ^^^  _^  _,,,^^ 

4-  ay-  ~  ay. 
6.  {a  +  ^)^  _  ,.^ 


2.  r/."  -  r^V 
3-  «V  -  X 


5-  («  -  dy  ~  c\ 
7.  :?^*  {x  -  ?/)^  _  a;*. 

9.     («  4- ^,)3  _|_  ^3^ 

II.  a^-^  +  y". 

13-  «"  +  04wV. 

15.  .'^•'  +  ^^ 

17.   «'  +  216. 

19-  ^*  +  ^*. 

21.   8a'  ~  27b\ 

23.   a;''  +  l. 

25-   «'  +  «i'. 

27.   ab*  ~  b\ 

29.   32^'"  +  1. 

31.  2.t'+16. 

33.   {x-\-yy  -(^x-yY 

35.   (a:  +  I/)*  _  (a..  _  ^)* 


8.  :?'  +  8xij\ 
10.  rt  («  4-  ly  ^  ^.^ 

12.  a'  Jr  (ib\ 

14.  /«"  +  ^Am'x\ 

^6.  r.'^"  +  8. 

18.  64.?;'-[-  12oc'. 

20.  :2-'  —  a\ 

22.  fJ4m'  —  8/^'. 

24.  Ua'-\-h\ 

26.  «  -  2r«\ 

28.  rt'  _  243. 
30.  KJr/'  —  ^^ 
32.  27«^-f  8ar'. 
34. 


36.   1 


(^  + ;/)"  4-  (.r  -  //)' 


(f  +  0^.:i;  -_  x' 


Factoring  Trinomials,     A  trinomial  of  the 


form 


X 


+  ^ix  4-  /; 


cuii  always  be 
sum  is  «  and  wh 
numbers,  the  tr 


factored 
lose  p 
nomi 


whe 


nwG  can  find  two  numbers  wh 


whicli 


■odiict  is  b.     For  if 
i  is 

+  (>/*  4-  n)  X  4-  7nn, 


ose 


?«  and  n  are  these 


is  0 


<\\\ii\  to 


(^  4-  «0  (-^  4-  «)• 


470 


FA  CTOIUNG. 


\         » 


Factor: 

I.  a-^  -\-  {a  +  i)  a:  +  ah.     Ana.  {x  -\-  a)  {x  -\-  h). 


3- 

5- 

7- 

9- 
II. 

IS- 


;'/' -!-  ^11  +  2.     Ans.  (?y  +  1)  (y  +  2). 

f   f-  4y  +  ;3.  4.   ^'  -f  5:r  +  4. 

'/i'  +  bn  +  G.  6.  w'  -j-  G?i  4-  8. 

a''  -h  '''^^  +  10-  8.  a'  +  8r?  +  12. 

wi*  +  7m  4-  VZ.  lo.  w."  -t-  8w  +  15. 

a:'  +  Tx'  +  lOrr.  12.  ?/^  +  G//^  +  8?/. 

.T^  +  Ix'  -f  12:r'  14.  a'  -^  8rt'  +  15rt'. 

a:^  -f-  19.c'  +  88.  16.  a'  +  12^'  +  35. 


17.  x'»H-  9a;" +  20. 


18.  y»+5^^"  +  6. 


19. 


x"^  -\-  {m  —  n)x  —  mn.     Ans.  {x  -f  tn)  {x  —  n). 


From  this  last  example  it  is  seen  that  Avheii  the  quantitit's 
m  and  n  have  opposite  signs  the  last  term  of  the  trinomial 
will  be  negative,  while  the  middle  term  will  have  the  sign  (if 
the  greater  of  those  quantities,  being  equal  to  their  algebraic 
sum  or  nuraerical  difference. 


20.  X 


X  -  Q.     Ans.  {x  -  3)  (x  -r  2) 

21.  x'  -  x"  -  12. 

23.  a'  -\-a-  30. 

25.  7/i'  +  2m  —  8. 

27.  n""  ~  3^'  -  40. 

29.  a;'  +  (2rt  —  35)  a;  —  6a5. 

31.  a;*  -f-  ax*  —  6a'. 


22.  y  -  2/  -  15. 
24.  a^  —  a  —  30. 
26.  wi"  —  2wi  —  8. 
28.  m*  +  3m'  -  40. 
30.  x^  —  3«a:  —  4rt'. 
32.  x"--  —  45a;"  -  125'. 


If  the  quantities  m  ?ii,d  7i  are  both  negative,  the  sum 
m-\-n  will  be  negative  and  the  product  positive,  because 


[x  —  m)  (x  —  it)  =  X*  —  [m  +  n)  x  -f-  mn. 


35- 

37- 

39- 
41. 

43- 
45- 
47- 


rr'  —  {a  -\-h)x  -\-  ah. 
,f  -  37/  +  2. 
x"  -  13a;  +  40. 
ax"^  —  ^ax  -f  2t7. 
?7i'a;'  —  5???  a;  +  4. 
??^^r'  —  3m. -c  +  2. 
aV  -  7rt^T  +  12^7. 
n'lf  -  Wf  +  10?j'. 


34.  y^  -by  -^  6. 

36.  a;'  -  Ix  +  10. 

38.  .9:'  -.  8.T  +  15. 

40.  ax*  —  Qax.^  -\-  8a. 

42.  7n'^x^  —  5ma;  +  4 

44.  m'.'/:'  —  4m.?;  +  4. 

46.  m\T'  -  7?>?V  +  12w'. 

48.  ?•"//'  -  IrY  +  12?-'7/'. 


FACT0RI2iQ. 


471 


en  the  quantitit's 

of  the  trinoniiul 

have  the  sign  of 

to  their  algebraic 


egative,  tlie  sum 
tive,  because 


111  the  following  exercises  trinomials  of  all  the  precedincr 
classes  are  contained:  ° 


x"  -\- 


.?;'  — 
a  c   - 


a'  — 

.3 


I 

3 

5 
7 
9- 

II.    .7-'  + 

13.  a'  + 

15.  a'  + 

17.  :i-^  + 

19.  X*  — 

21.  a:'*  — 

23.  (a  + 

24.  .'^'  -}- 
26.  .t' 


IO2:  +  ?4. 
Sx  -  30. 
7.^;'  +  12. 
-  IQabc  4-  3!)/*-^ 
12a +  20. 
^x  -  32. 
«'  -  132. 
17«'  -  390. 
X  -  72. 
39.r  +  108. 
7.7;  -  60. 

*r-llc(«  +  ^) 
4.r  -  77. 
14.«  +  48. 


2.  .t'  —  6.?^  -f-  8. 

4.  :c*  +  3.c^  4-  2. 

6.  2:^'  -  27.17/  -f  2G. 

8.  a'b-  —  24^f'(^;?;  -f  143.r'. 

10.  .T-'  +  50.r  +  49. 

12.  a'  -  ^ui  ~  18. 

14.  r?^Z»V  +  Oa'b'c'  -  22. 

16.  a'  ~  7a   i-  12. 

18.  x'  ~  I2x  +  27. 


20. 

22. 

-h  306-\ 

25. 

27. 


.^-   -  ;*•  -  12. 
lox'  -  17x'  +  4. 

x'  -{-ex~  135. 
2;'  -f  122:  +  35. 


I. 

3- 

5- 
7- 

9- 

II. 

15- 

'7- 

19. 
21. 

23- 

27. 


Miscellaneous  Exercises  in  Factorinj?. 

ax"  -  2bx  -f  ex.  2.  a.T  -  2Z».r  +  dn/  -  ^.r  +  2//. 

«':i-  -  2c^/f/  -  4.T  -  y  -f.  a;.   4.   a*".?  -f-  a2»»^'  _  'S(f^\v\ 


ax'-  2bx'-}-  cx\ 
ex'  -  abx'''  ~  2y  +  Tiaif. 
'Zc'x'if  -  2-''?/'«  +  dx'y\ 
{x^  -  4). 


,^,Vy'  -  a'b'  + 


a.'*' 


6.  i,^''—  2iqx^  -\- px"". 

8.  «6\r?/  +  2xy  -  3x\fj\ 

10.  42-'^  —  3.T'*?/"'4-  2.<;"'_y'. 

12.  (.-r'  —  9.r). 

14.  4.r'  -  12.7:'^  + 0//'. 

16.  «V  _  2/^ 

1 8.  9rt»  -  1. 


4rt'  +  1  -  4a. 
.r'  +  2.2:>  +  xy\ 


20.   a:^'"  -f  a;'"  +  ^. 

22.   16a>"  —  1. 

25aV  -  30a.<y  +  9x'y\     24.   12a^  -  3Ga'2:y  +  273;'y\ 

-:!''_  -5  4.  i_  .    a'         a  1 

4        3?/  "^  9?/^' 


4//       6//^  +  187* 


.a'         a     , 

c        (ib  ^  18//" 


28.  a' 


i 


'I 


472 


FACTOEIIiG. 


»        ( 


m 


!r 


f 


29.  2ia'b  -  72a  y  +  54a*'.      30.  x'*  -  y'\ 

31-  (4-171).  3--  mV  +  2mV  +  mV. 

33.  ^SOaV/^"  +  lOSaJV-'^/  +  ^'c'cT. 

34.  12 la"  -  286a'^'  +  U%\    35.  98a'Z»'  -  bC>abx  +  8a;'. 


36.  Ux'  4- 8^'  + a;* 


37-   H^'  -  2/' 


39- 


a 


a  +  1. 


40.  Uab  {dab'  -  lObc)  +  20c  (5c  -  4aZ»'). 

42.  2^xy  -  13lxyz\ 

43-   («  -  l^y  -  («  +  ^Y-  44.  (2a  -  /;)^  -  (a  +  2/>V. 

(a  +  by 


41.  —  —  Sax 


I 


45 


46.  (a  +  by  -  (a'  -2aZ»  +  h'). 


4  (a  -  Z»)'' 

47.  (a  -  Z*)"  -  (4a'  -  12a6  +  9^*'). 

48.  (a  +  by  -  (4a'  +  12a6  +  %'). 

50.  a'  —  2a7/"  -h  f/''". 

52.  a'/^'  -  2ac^»  +  c'. 
54.  IGa'  -  4c'. 

56.   16a:'  -  24a;'  +  9a;. 


U  5 

49.  -.-  -  ^  +  -.- 

51.    a;2m  _4^^.»H^yn_^  4^/2n 
^2»  ;,,2n 


53.  x-'~y 

9a^'r"' 
55 


ICr'y 


*  25^y         4aV' 
57.  4a;'  —  4.t'  +  a;*. 

59.  (2a;  +  ^)'- 4  (a;  +  7/)' 


58.  4a'i'  -  (a'  +  ^''  -  c')' 


Products  of  Two  Binomials. 

We  have 

(a  +  b)  {x-\-y)=  ax  +  bx  +  a?/  +  ^y. 

Hence  a  polynomial  of  four  terms  may  sometimes  be  ex 
pressed  as  a  j)roduct  of  two  binomial  factors.  We  can  do  tliis 
when,  two  terms  of  the  polynomial  {ax  -\-  bx  for  exanipk) 
l)eing  divided  by  a  common  factor  {x),  and  the  two  remain- 
ing terms  by  a  common  factor  {y),  the  quotients  are  equal. 
We  can  thus  factor  the  following: 

1.  ax  —  bx  -\-  ay  —  by.     Ans.  {a  —  b)  {x  -f-  y). 

2.  ax  -\-  bx  —  ay  —  by.  3.  ax  —  bx  —  ay  -{■  by. 


DIVmON  BY  POLYJSOMIALS. 

4.  w'  +  mw  +  w'  +  71.  5.  mn  -  m'  +  ?i'  -  n'm. 

6.   1  -f  «  -f-  «'  4-  rt'.  7.   1  _  3,  __  ^>  _^  ^3^ 

8.   l-^-x-x"-  x\  9.  «3  _^  ^,^  _^  ^^.,  _^  ^,^ 

10.  {a  ~  nx)  {a  +  nx)  ~  {71  -  ax)  {71  +  ax). 

11.  b'-  Wx  +  ^»^'  -  '6x\         12.  a'  -f-  rt'  -  r/^  -  a' 

13.  wi"  -  3m^  +  m^  -  3///'.       ,4.  771^  +  3^/?/  -  m^  _'  3m 


473 


hi 


ometimes  be  ex 
We  can  do  tliis 
hx  for  exainpU) 
the  two  remain- 
tients  are  equal 


Division  by  Polynomials. 

§97. 

1.  «'  +  4:ax  4-  4.r'  ~  a-^2x. 

2.  6r<*  -  r,//"  ^^>«'  _  :>^,». 

3.  ^^  -  3«V;»  -f  'da'b'  -b'^a'-  3a^'b  +  3«^'  -  b' 

4.  «=>  _  9rt'  4-  27«  -  27  ^  ^r  -  3. 

5.  48a'  -  7Grt'(^  -  eiab'  +  105^*^  4-  2a  -  35 

6.  ia'  +  a*+  ta  +  f  -f-  i-a  4-  1. 

7.  33«'Z/'  -  77a'Z.^  4-  121^7/ -  3^"^^  -  7ab'  4-  l]a5» 

8.  lOOrt^  -  440a^Z*  4-  235a'b'  -  dOa'b'  -f-  5a'  -  2a'b 

9.  37a'^''  -  26a'b  +  3a*  -  14a5'  --  3a^  -  6ab  4-  2b\ 


10. 


..nt  +  l 


4-  a-my  ^xij'^-^y 


m  +  1 


™fi 


+  !/' 


I  I.    «*«  4.  f^2»^2H  _^  ^4»  _^  ^2n  _^  ^„^n  _|_  ^ 

12.  10a*  -  27a'b  4-  34a^<5*'  -  ISai'  -  8b'  ~  20"  -  Zab  4-  W 

13.  4.rl  —  3%i  —  7/  -^  xi  —  yi. 

14.  8a*  —  Ga}  -\-  ai  -i-  2ai  —  a  A. 

15.  9a-24-12a-^4-4-4-3a-i4-2. 

16.  4x  -  lO.ti  -  Q>2xl  -  30.ri  ~  2xi  4-  5. 


17.  x'y-''-{-x-^if  ~ 

18.  x^^'  —  f 


X    'y-\-xy 


X" 


y 


19.  4.7;'  4-f?.  -  Tox'  4.  58:^;*  -  7O2;'  -  23:r  -  fj:^;'  -  5:r  4-  2  - 


20.   .r' 


-  ^  «"^'  + 


:x\ 


2a' 


c/ 


+  ,T  -^./'^  —  2aa:4-  -' 


(f 


21, 


(«'  -  2a/;  +  b'-c'){a  +  b-^c)~a-b- 
{ax  4-  %)^  4-  {ay  -  i:r)'  -f-  a'  4-  /*' 


23.   12.7;*  -  14.r'  -  ll:c'  4-  19.r  -  G  -^  3: 


oa; 


+  2. 


24.  W¥  ~  35*  4-  llab'  4-  12a*  -  34a'5  -  5^  4-  Ga^  -  5a6. 

25.  Gabc'  -  Wc'  4-  4a'5'  -  a^c'  -4-  35c  -  ac  4-  2a5. 

26.  25c  -  1  4-  a'  +  2c  -  5^  -  25  -  c'  -  a  +  c  -  1  4-  J. 


in 


474 


DIVISION  BY  POLYNOMIALS. 


U  i 


16 


2x 


4 


z 
2' 


27.  3^'- 82:2- ^-7/  +  -  ^     .    g 

28.  a;   -y   -z   _-^  +  — -  +  -^--.^-3.  +  -^. 

29.  !:>  {x  -  yY  -  3.6-  iy-z)-  2y  {x  +  z)  -  20z  {y  +  3z) 

--  5  (;r  +  2z)  —  Sy. 

30.  {Ax'  -  9/)  (8./;'  -  27y')  ^  {2x  -  3//)\ 

31.  12  +  S2a''  +  100^^'  -  lOa'  -  ll2a'  -  38  ^  3  -  ba  +  7a\ 

32.  «,'  {b  +  r;)  +  ^'^  (>t  ~  r)  +  c'"'  {a  -  b) -{- abc -^  a -{- b -^  c. 

33.  «'  +  Z>'  +  6'-'  —  '^abc  -^  a  ^  b  -{-  c. 

34.  :c'  —  (a  +  ;;)  /'  -f  ('7  +  ^Jf^)  '^'  —  crq  ^  x  —  a. 

35.  rt'  -  ]3f/.'  +  30  ~  a"'  +  6a  +  6. 

36.  .T*  +  -^'y  + .'/'  ~=~  •''*'  ~  '^7/  +  //"• 

37.  3rt'  -  8rt'^^  +  3r7V  +  5^^  -  bv-'  -=-  a'  -  b\ 

38.  y'  -  ^y\v'  +  3?/'a-^  -  x'  ~  y'  -  3//1?;  +  3?/^'  -  :r^ 

39.  IGrt'a:;"  —  7rt6c  —  c'  —  4:a'bx  —  6rtV/' -f  Br^t'.c  -^  8^2;  —  Gab  —  c. 

40.  a:"  +  {a'  -  2b')  x'  -  {a'  -  b')  x'  -  a'  -  2a*b'  -  a'b' 

-^x'  -  a'  -  b\ 

41.  6  {x'^tf)  +  (18.r//  -  4)  (.T+?/)  -  8  (a:'  4-  2/')  -  l(jxy  -  120 

-  x'  +  f/"  +  2a-  (1  +  ^)  +  %  +  G. 


a' 


-  J'  -T-  «?  -  5t. 


43.  a'b''^  +  2rt6'Z>"'  +  '*  +  2axb'^  -f  c'6"*  +  2c.rh''  H-  ^t' 


^  ab""  +  6'J"  +  .'T 

44.  (.t'  -  .?/')  (r^;^  -  f)  ^x'  +  /. 

45.  20a''V  -  208rt^6'*'  -  121a''b''  +  132a"^»*  +  245a"6" 


46.   1  +  34  a:**  —  20.r'  +  20.t'  -  4^:'  -[-  12.1- 


3l2;« 


-^  2:c  +  4rc'  -  3.c'  +  1. 


47.  22;^"  —  e.r'^y"  +  ea;^/'"  —  2?/^"  -^  2;"  -  ?/' 

48.  a{a-l)x'  -{-  [a*  +  2«  -  2)  a;"  +  (3 


a 


a 


V- 


rt 


-r-  ^a;"  —  22;  —  fl^' 


49.  «^?y'  —  b  {a'  -{■  b)  y  -{-  aV  -r-  ^?/  —  b. 

50.  (rt  +  ^)  (a  +  c)  —  (a  +  ^)  {dArC)~-a  —  d. 

51.  x'  +  {4:ab  -  b')  x  -  {a  -  2b)  {a'  +  3b'')  ■^x-a-\-2b. 

52.  x'  —  y~^  —  x^  -\-  y~^. 

53.  i  -  C^'  +  21z'  -^ii  -h  2^  +  dz\ 


54.  a^"*  -  ""b'^Pc  —  a^"'  +  "  -  ^Z»^  -^r"  -f  a  "  "^  "  ^c"*  +  «'"• " "J^''  +  V 


—  a 


2m  +  2)1  — 17,3, .2)1  —  1 


+  /^ 


;)  +  1  .,m   t  n  —  1 


rt 


n/i-P-1 


4-5c 


n-1 


z  {y  +  3^) 

3  -  5r^  +  la\ 


■  ^ax  —  6ab  —  c. 
?  -  a'U 
^x'  -  a'  -  h\ 
')  -  Uxy  -  120 
-f2/)  +  2;y  +  6. 


-I-  ^'^ 

f  245a"6'' 
IGa^*"  +  \la'b\ 

-  4rc'  -  32;'  +  1. 

a^)  X  —  a" 

-  ax^  —  2x  —  a\ 

d. 

-  X  —  a  -{-  '^h. 


FRACTIONS. 


475 


Fractions. 

§  108. 

Execute  the  following  multiplications  of  fmct ions  l,y  en(  iro 
quantities  by  dividing  the  denominators-  ^ 

(i  +  b 


"  a~b 


x'  -  4.y 


X  a-  b. 
XX-  2?/. 


w?' 


a'  -  b' 


""'  7^~l;^^''  +  ^' 
A    P"  +  (?' 


1  +X 


7)1   71' 

X  l-{-x. 


Execute   the  following   multiplications   by  dividing  the 
.lenominator  by  one  factor  of  the  nuUtiplier/when  d  nom 
nntor  and  multiplicator  have  a  commoli  di  isor  '^ndTen 
n;ultiplying  the  numerator  by  the  other  factor  rf^rernuU;. 


m 


a' 


~  2ab  +  y  ^  ^'  ~  ^' 


ler  is 


Here  tl,e  denominator  is  (a  -  by,  and  the  multipl 
{a  -  i)  («  +  J).     We   multiply  by  (o.  -  i)  by  dividing  the 
denommator,  and  by  «  +  J  by  multiplying  tl  numefator! 

TT^„«^  ^T -,     >  •    7n(a4-h'\ 


Hence  the  product 


is 


{a_±b) 
a  —  b 


h 


711 


-\-  2m7i  -j- 
h 


n' 


X  wi'  -  n' 


m' 
a  ~  a 


4:7nn  -f-  471 


~,  X  771'  ~  W 


4.  -5^—,  X  («  +  xy 


a'  ~  X 


a  ~  x 


«  +  a'^x  -  ax' 


3^  X  {a'  -  x')  {a  +  xy 


1 


rr-;-r-  X  a' 


ax  +  n'?/  +  bx  +  /^; 


¥' 


„  jj . 


t       t 


»         ♦ 


»• 


■■■I 

tp. 


476 


1 


ax  —  ny  -{-  bx  —  by 


FRACTIOUS. 


X  a'  +  h\ 


8. 


1 


w.r  —  nx  —  my  -f  vi^ 

a  — 


X  wio;  +  w*2/  —  ?i^  —  W7/. 


'•  «='  +  ^' 


X  rt'  +  Jir/^  +  h\ 


Execute  the  following  divisions  by  dividing  the  numerator 
by  us  many  factors  of  the  divisor  as  possible,  and  multiplying 
the  denominator  by  the  remaining  factors: 


ax 


X 


3 

5 

7 

9 
II 


-V-  ar.     Afis. 
mn  mnr 

ah       ^ 

7nn 


my 
2.  -—-  -T-  7nq. 

«'  +  ^^        J 
a  —  0 


m 


p  m^         4        4 
^         -i-m  —p. 


-ir-x^  —  y^. 


ex  -f-  cy 
ax  —  ay 

_____  ^  («  4.  ^6)  . 
ac  —  be 


6. 


^+2/ 


a:  —  2xy  -f-  //' 


i'. 


-4- a' 


//■ 


ca:  —  c,7/ 

8.  -^ — —-r-  -5-  (^'  —  y) 


10. 


Z»a:  +  Z*?/ 


a 


T-    rt'    -    b\ 


ax  -\-  bx 


-=-«'-  b\ 


Execute  the  following  indicated  multiplications  or  divi- 
sions, and  aggregate  each  product  or  quotient  into  a  single 
fraction : 


wi       nJ  \m       nj 


'■  (-  - 1)  -• 

\a       bJ  c 


m 


m 


5- 

7. 

9- 
II. 


a  -\-  b      a  —  bl  a 
m       711  \m       n 


1  +  x 

(1  +  ^ 
\        m 


+ 


1 


•T 


'^'-111 


i> 


V         1  —  ml  1  -f  m 

8.     ^(,;i»4.2  +  i). 


X 


a 


a  —  XI  X 


m 


g  the  numerator 
anil  multiplying 


a'- 

■b\ 

■h?/ 

■~x' 

-(..•- 

-yy- 

ilications  or  divi- 
lent  into  a  single 


FRACTIONS. 


477 


■+^)a-'")+-](^:-a. 

t  J.    {m  +  n)  (i  +   ^-)   -  (,;;  _  ,,^   (i  _  1] 


'^'  -.!i+^-^^+^- 


17       ^  1     . 


Wi 


i8. 


20. 


21. 


1 1_  _^  a-j-b 

(i-i-i      a  -h   '  a~b'     ''^' 


c'  +  /i 


i  + 


/^' 


c'  -h 

1  .  J 


4* 


l+;  +  ^.  +  ;:.-z  +  r, 


a;       a; 


a: 


X 


X 


Factor  tlw)  following  fractional  expressions: 


c  X 

a:         c 


3. 

5- 

9- 


a' 

4.   -,-  - 
a; 


y 


a  • 


3* 


X  X 


6    '±{P^J^) 

.0.^  +  54  +  42. 

r  r  r 


§110. 

Reduce  the  following  complex  fractions  to  simple  ones: 

c A 

h       c 


I. 


1  +  1 


1_ 


1^ 

b 


2. 


1  +  i* 


^1 


Pi';  I. 


478 


SIMPLE  EQUATIONS. 


i  '       • 


'I 


i'l 


iir:- 


% 

■  i 


i 


f 

i 


'  -I-  '^  +  ^ 


1 


:.  +  77  + 


ft 


7'  -f-  S 


'     +     ' 


T  -{-  s   ■   r  —  s 
r  -f-  .9 


1  + 


ir. 


7*  —  .«? 


13- 


^^ ^ 


1  4-  m 
4. ^-. 


6. 


8. 


m 


lO. 


12. 


14. 


1 

r 

r  +  - 
r 

1 

7W 

+  i+,^ 

1 

11 

mnp 

1 

1 

m 

?i 

w' 

Vi 

m 

Ecxtiatioiis  of  the  First  Degrree  with  One 

Unknown  Quantity,  x, 

§139. 


2. 


X  —  m      x-\-  n 
n  m 

(rx  .   bx   ,    ex 


+  2  =  0. 


+  ;;;:  +  .-^  =  «^'  +  ^'  +  ^^ 


be    '   ac    '   ab 


3.   (a  -  a:)  (5  +  a;)  =  a;  (5  —  x). 

bx  .   ax        . 
4-  —  +  -.-  =  «'  +  ^'. 


a 


^-  ^^■'^-^;  +  ^'^  =  T  +  («  +  ^)^^'-<^' 


svith  One 


ONE   UNKNOWN  QUAyTlTY. 


(y. 

7. 
8. 

9. 


,   he    ,   (i'x 


V 


X 


a 


I      a^b 


=  1. 


ox    ,    rx    ,    (fx 

-  ad  -{.~nc~~  ex  =  ^ac  +  2ab  -  Gcx. 


X 


_  1 


dx 


I  T. 
12. 

14. 

15- 


.'•  X 

^n^x  _  .  __  0 
^-'  x' 


+  SrtZ"  =  0. 


c  —  a  -\- 
1 


m  {a  —  x) 
~:ia'+x~ 
X  -j-  a 


X  +  rt       X  ~  a       a'  —  x^ 


7.?-' 


(]x 


n  f  1 


17. 


,7;  -   1 


X-  4 


-f.r"       3.7:"+r).?-«  +  ' 


cc  +  l 


a; 


6 


.7: 


•> 


=  0. 


a 


+ 


a 


J7,a 


+  *   '   (a  +  b) 


rT3  + 


(2(^  4-  ^)  b'x 
a  {a  +  Z*)' 


OCX  H . 

a 


a'c 


18.   {a  +  2:)  (^>  +  .7;)  -a{b-\-  c)  =  -~  + 


X' 


19. 


20. 


X  —  a  ,   a:  —  5 


a 


+ 


+ 


a;  ■—  c 


+  3  =  0. 


x-^  a      x-{-a^   ,   ^'-\-a^      x-\-  a 


a 


rr 


--  + 


a 


a 


a 


21.   {ax  +  (5i)  (Z>.r  —  a)  -  {ax  -  Z»)  (^.r  -\- a)  =  a  H-  /^. 


22.  rt'^  — 


a  -\-  x 


~a¥  - 


b-{-x 


a 


m  —  71 


+ 


m  ~  n 

X 


m  -\-  n 


+ 


m-\-n 


24. 


_        P 


{m^ny       m-\-n      2  {m  +  71)' 


f 


P 


479 


^  ' 


M      ' 


480 


SIM  PL  /'J  KQ  UA  riOSS. 


f        I 


X 

H 


,        X  —  tn 

^  1  —  nix 

p  4-  X         nx 
26.  in  —    -. —  =  —  -. m. 


<l  -\-  f      q  -f-  X 


1 


"  + 


1 


1 


ab  —  ax       be  ■—  b.v       ac  —  ax' 

(ill''  —  ;<■')  m 


2cS.  {ill  4-  ny  =  3//i'  -f-  n' 


X 


(      I      \i      n    1  ,     9   ,   ni{)n''-n^) 
29.   {ill  -f-  ;i)''  =  .inr  -\-  7i  -] - 


„      b'-c'       b<'(b-\-c) 

30.   b'  =    r ^ — —^' 

b  —  c  X 


VI 


7n 


31- 


l-\-x       1 


X 


n 


+ 


71 


=  1. 


a; 


/?i  -| — 
32. =  2m. 


14-^      1  —  « 
.r  —  re 


1  + 


z:-,- 


X  -\-  a 


m 


1  + 


1  - 


X  —  a 


=  A. 


34. 


X 

X  —  a 
X  -{-  a 


"0  -\-  a 


1  + 


X  -{-  a 
X  —  a 


=  Vi. 


Eciujitions  of  the  First  Decree  with  Two 
Unknown  Quantities. 

§§137-140. 

Spx-{-  qy  =  a. 
I    X  —    y  =  b. 


j  mx  —  ny  =  0. 
(    X  -{-   y  =  a. 


2. 


3- 

5- 


\ax-\-  a^y  =  ap. 
a  b 


b  +  //         3«  +  ^ 
ax  -\-  2by  =  d. 


7.    i 


X       y 
a       b  _ 

X       y  ~   ' 


6. 


8. 


y    ^3 

x-\-y       3  • 
-^=3. 

bx-\-  y  =  a. 
x  -|-  by  =  2a. 


I  r, 


TWO   UNKNOWN  qVAyTlTIK!^. 

b 


481 


lo.   I<     y"^' 


i;v  ax-{-b  =  my  -\-d  =  c, 

15.   m{x-^y)  =  n{x~y)  =r. 

'     X  y      _      1 

16     J  «  +  *  ~  (^  -b  "  ^Tfl* 


II 
h  (.r  -  ,/)  4-  a  {,f  -  /,) 

X  -  a  \  y  -  h  =z  h  ;  ,t. 
14.  mz  =  ny  —  p  =:  X  -}-  //y. 


1 


+ 


1 


17. 


1-x-^y    ■    1  ~x-y       3* 

1 ]___  ^  4 

1  —  ;i;  4-  //       1  _  ;/:  _  ^y  —  3  • 


•«l 


'   I 


vith  Two 


EquatioiiH    with    Tliree    or    More    Unknown 

Quantities. 


7. 


m:c  —.  ny. 

py  =qz. 

x-\-y  =  a. 
y  -\-z  =  2a. 
z -i-  X  =  3a -\- b, 

Ux  =  j^y. 

]  y  =  iz. 
ip  =-^x-\-l. 

X  —  z  —  am. 
y  -\-  z  =  bm. 
X  —  y  =  cm. 

X  =  y  ~  2z. 
y  =i''Sz  —  2x. 
z=y-^l. 


2. 


6. 


8. 


10. 


m  _  n  __  p 

X       y  ~  z' 

X  -\-  y  -{-  z  =z  s. 

X -{- y -{- z  =  30. 
Sx  +  4v  -\-2z-  50. 
27:?:  -\-9y-\-3z=  64. 


'x-\-y  -\-  z  = 
X  -j-  y  =  bu. 

au. 

X 

[y 

nx  -\-  y  -\-  z 

X  -\-  ny  -\-  z 

—  a, 
=.b. 

•r  -  f-  //  +  nz 

=  c. 

n 


482 


SIMPLE  EQUATIONS. 


*3 


I  I, 


T^Z- 


15- 


17. 


X  +  2?/  =    8. 

y  -\-  iz  =  U. 

z  ■\-  u    =    8. 

JO  -\-  X    =    4. 


(  ax  -\-  by  -\-  cz  =  d. 
\  a^x  -f-  b'^y  -\-  c'z  ■=.  iV, 
[  a'x  +  b'y  +  c'z  =  d\ 


(  2x  +  3^  +  5^  =  r;7. 
\  -  2.C  +  :3?/  +  4^;  =  35. 
(  -  2a;  -  3?/  +  5:2  =  13. 


'  X  +  y  ~  ct' 
y  Vz^b. 

z  -\-  u  =  c. 
u  —  X  =.  d. 


rx  y 

a       a  —  r 


12. 


^ 


+  -----  -  1. 

a  —  s 

:>'  V  Z 

b   ^  b  -r^d-  S 


Si 


14.     i 


16. 


V  z 

.T       y       « 
X       z        b 

i  +  Ui. 

(3.r+    2/+    2  =  3. 
j    a;  4-  4?/  H-    ^  =  4. 

(    ^  +    2/  +  ^^  =  ^' 


r=  1. 


X 


18.    ^ 


+ 
+ 


b  -\-  c      c  —  a 

y     .     ^ 

c  -\-  a 

z 


^^  =  a-{-b. 


a  —  b 

X 


a  -\-  b      b  —  c 


=  b  +  c. 
=  c  -{-  a. 


19.    S 


f  bx  -^ay  _        a  —  b 
b-c 


c 

cy  -- 

bz 

a 
az  -- 

ex 

{c—a){b—ay 
c  —  a 

{a-b){c^Y 


:t| 


*-■•: 


PROBLEMS     LEADING    TO     EQUATIONS    WITH     ONE 
UNKNOWN     QUANTITY.* 

1.  A  capitalist  earned  4  per  cent  interest  from  |-of  his  in- 
vestment, and  5  per  cent  from  the  remaining  ^,  making  a 
total  annual  interest  of  $2940.    What  was  the  amount  invested  ? 

2.  What  quantities  must  be  added  to  each  term  of  the 

777, 
fraction  —  that  it  may  take  the  following  series  of  values: 

*  Although  only  oiip  unknown  quantity  is  renlly  necessary  in  these 
problems,  the  student  maj'  often  find  it  convenient  to  use  two  or  more. 


,:.V^. 


moBLEys. 


483 


1. 


r 


a 

— 

s 

b 

z 

s 

zrz 

i. 

z 



1. 

S\ 


z  = 

3. 

z  = 

4. 

bz  = 

5. 

y    . 

—  a 

=  a  -{-  b. 

z 
=1 

=  b  +  c. 

X 

-  c 

=1  c  -{-  a. 

ITH     ONE 

m|-of  his  in- 
\,  making  a 
unt  invested  ? 
term  of  the 

of  values: 


cessary  in  these 
se  two  or  more. 


[c)h 


y>)  .y  ".."' 


n 


('')"■ 


What  quantities  must  bo  sul)tracte(l  from  each  term  to  pro- 
duce tlic  same  results  ?  Explain  the  relation  betwoen  the 
answers  in  the  two  cases. 

3.  A  man  is  40  years  old,  and  his  wife  is  36.  In  how  many 
years  will  the  sum  of  their  ages  be  si'  E:?i:j)lain  t'lc  results 
when  we  put,  in  succession, 

5  =  100;         s  =  70;         and         s  =  50. 

4.  A  railway  train  passed  a  station  at' the  speed  of  ?/i  miles 
an  hour.  Then  k  hours  later  another  i)assed  in  the  same 
direction,  going  71  miles  an  hour.  Supposing  the  speeds 
uniform,  at  what  distance  and  at  wliat  time  did  they  meet? 
Explain  the  relation  of  the  answers  when  vi  >  n  and  when 
)n  <  II. 

5.  If,  in  the  preceding  problem,  the  second  train  went  in 
the  opposite  direction,  what  would  the  answer  be?  Explain 
the  relation  between  the  answers. 

6.  A  ship  sailed  from  port  with  a  speed  k  knots  per  hour. 
In  h  hours  after  sailing  she  was  followed  by  a  steamer,  who 
overtook  her  in  11  iiours.     Y/hat  was  the  speed  of  the  steamer? 

7.  An  oarsman  who  pulls  6  miles  an  hour  rows  from  his 
house  down  a  river  whose  current  is  2  mi.les  an  hour,  a^id  re- 
turning gets  back  3  hours  after  he  started.     How  far  did  he 


go? 


8.  On  the  same  stream  one  rower  pulling  6  miles  pn  hour 
going  down  stream,  and  another  pulling  7  miles  ar.  hour 
going  up  stream,  started  out  at  the  same  moment;  but  the 
t<tarting-point  of  the  second  was  5  miles  below  that  of  the 
first.     At  what  point  and  in  what  time  did  thev  meet? 

9.  A  steamer  goes  down  the  Rhine  from  Mayence  to  Co- 
logne, 117  miles,  in  8i-  hours,  but  requires  14  hours  for  the 
return  journey.     What  is  the  speed  of  the  current? 

10.  On  an  ocean  the  crests  of  the  waves  are  -^jy  of  a  mile 
apart,  and  are  moving  at  the  rate  of  40  miles  an  hour.  If  a 
ship  steams  15  miles  an  hour,  hoAv  many  times  an  hour  will 
she  pitch  when  going  with  the  waves,  and  how  many  times 
wheii  going  against  them? 


484 


SIMPLE  Eq  UA TIONS. 


iH'fc 


1 

> 

»«i 


11.  A  number  is  divided  into  three  parts,  of  which  one  is 
30  less  than  a  half,  a  second  10  less  than  a  third,  and  the  re- 
maining part  8  greater  than  a  fourth.  Find  tlie  number  and 
the  thrr      arts. 

12.  i.om  a  line  was  taken  \  its  length  and  2  fcec  more, 
and  from  what  was  lef t  ^  its  length  and  2  feet  more,  leaving 
^  the  whole  line  and  2  feet  more.  What  was  the  lengtli  of 
the  line? 

13.  A  team  performed  a  journey  in  8  hours,  going  one 
third  the  way  at  the  rate  of  25  miles  an  hour,  and  the  remain- 
ins^  two  thirds  at  the  rate  of  40  miles  an  hour.  What  was  the 
distance? 

14.  A  grocer  has  60  pounds  of  tea  worth  75  cents  a  pound, 
formed  by  mixing  one  kind  worth  80  cents  a  pound  >vith 
another  worth  50  cents  a  pound.  How  many  pounds  of  each 
kind  were  in  the  mixture? 

I-.  Divide  a  line  of  lengtli  I  so  that  |  of  one  part  shall  be 
equal  to  \  of  the  other  part. 

16.  A  man  is  6  years  older  than  his  wife.  Ten  years 
hence  the  sum  of  their  ages  will  be  7  times  the  age  of  the  wife 
14  years  ago.     Wiiat  are  their  ages? 

17.  A  man  whc  must  be  back  in  1  hour  starts  in  a  coach 
going  VI  miles  an  hour,  and  walks  back  at  the  rate  of  n  miles 
an  hour.     How  far  can  he  go  and  be  back  in  time.? 

18.  The  earth  performs  a  revolution  round  the  sun  in  1 
year;  Mai:,  in  1-^-  years.  What  is  the  mean  interval  between 
conjunctions;  that  is,  between  the  times  at  which  the  earth 
passes  Mars? 

19.  The  periodic  time  of  Jupiter  is  11^  years;  of  Saturn, 
201  years.  At  what  intervals  will  the  earth  be  in  conjunction 
■with  each  of  them,  and  at  what  intervals  will  they  be  in  con- 
juaction  with  each  other? 

20.  Two  persons,  A  and  B,  were  mounting  a  tower,  B  be- 
ing always  24  steps  behind  A.  When  A  was  half  way  up  lie 
said  to  B,  *'  When  I  reach  the  top,  you  will  be  8  times  as  higli 
as  you  are  noAv."     What  was  the  height  of  the  tower? 

21.  The  circumference  of  the  front  wheels  of  a  carriage  is 
9  feet;  of  the  hind  wheels,  12  feet.     How  far  has  the  carriage 


PliOBLL'MS. 


4^5 


which  one  is 
i,  and  the  re- 
3  number  and 

1  2  fcec  more, 
more,  leaving 
.  the  length  of 

urs,  going  one 

[id  the  remain- 

What  was  the 

cents  a  pound, 

a  pound  >vith 

pounds  of  each 

Qe  part  shall  be 

:e.      Ten  years 
age  of  the  wife 

:arts  in  a  coach 
rate  of  n  miles 
:ime»? 

nd  the  sun  in  1 
nterval  between 
which  the  earth 

?ars;  of  Saturn, 
in  conjunction 
they  be  in  con- 

g  a  tower,  B  he- 
half  way  up  lie 
8  times  as  high 

le  tower? 
of  a  carriage  is 
has  the  carriage 


I 


driven  when  tlic  front  wheels  have  made  rn  turns  more  than 
the  hind  wheels? 

22.  The  members  of  a  club  have  to  raise  a  certain  sum  of 
money.  If  each  member  contributes  $2,  there  will  l)c  ^)IS  too 
much;  if  $1.25,  there  will  be  $32  too  little.  How  many  mem- 
bers are  there,  and  what  is  the  amount  to  be  raised? 

23.  If  a  dealer  sells  a  piece  of  cloth  at  vi  cents  a  yard,  ho 
gains  d  dollars;  if  at  71  cents  a  yard,  he  loses  c  dollars.  What 
is  the  ler.gth  of  the  piece,  and  the  purchasing  price  per 
yard? 

24.  A  merchant  by  the  profits  of  trade  increases  his  capi- 
tal each  year  by  20  per  cent  of  the  aiiount  at  tlie  beginning, 
but  takes  out  $1000  at  the  end  of  each  year  for  his  board.  At 
the  end  of  the  third  year  he  has  increased  his  capital  by  $200 
more  than  -^  of  its  original  amount.  With  what  amount  did 
he  start? 

25.  A  boat  which  steams  12  miles  an  hour  makes  her  trip 
in  3  kours  going  down  stream,  and  in  5  hours  going  up  stream. 
Wliat  is  the  speed  of  the  current  and  the  length  of  the  trip? 

26.  A  number  is  increased  by  ??,  and  the  sum  multiplied 
by  ;r,  this  product  is  then  increased  by  n,  and  the  sum  multi- 
plied by  n,  with  tlie  result  2«'.     Wluit  is  tlie  number? 

27.  A  number  is  diminished  by  v.  and  the  remainder  multi- 
plied by  n  ;  the  same  operation  is  repeated  on  the  product, 
and  again  repeated  on  the  second  product,  with  the  result 
—  ?i'.     What  is  the  number? 

28.  What  number  is  that  whose  fourth  part  exceeds  its 
sixth  part  by  2? 

29.  If  you  add  4  to  a  certain  number,  the  sum  is  2  less 
tlian  twice  the  number.     What  is  it? 

30.  Divide  $.";20  among  tliree  people  so  that  the  first  may 
liave  $20  less  than  the  second,  and  the  second  $10  more  than 
one  fourtli  the  share  of  the  third.     What  must  each  receive? 

31.  Divide  c  dolLars  among  three  people  so  that  the  first 
may  have  a  dollars  less  than  the  second,  and  the  second  m 
dolhirs  more  than  one  fourth  the  share  of  the  third.  What 
nmst  each  receive? 

32.  A  left  a  certain  town  at  G  miles  an  hour,  and  in  8 


SIMPL  E  EQ  UA  TIONS. 


\'>       I 


!  ! 


lb 


hours  after  was  followed  by  C  at  8  miles  rsr  hour.     In  how 
many  hours  did  C  overtake  him? 

T,T,.  A  left  a  certain  town  at  b  miles  an  hour,  and  in  n 
hours  after  was  followed  by  D  at  c  miles  per  hour.  In  how 
many  hours  did  D  overtake  him? 

34.  A  farmer  said,  if  he  had  5  more  sheep,  and  sold  them 
at  $4  each,  he  would  have  5  times  as  many  dollars  us  he  now 
has  sheep.     IIow  many  sheep  has  he? 

35.  A  farmer  said,  if  he  had  a  more  sheep,  and  sold  them 
all  at  n  dollars  each,  he  would  have  c  limes  as  many  dollars  as 
he  now  has  sheep.     How  many  sheep  has  he? 

36.  If  you  divide  my  age  10  years  hence  by  my  age  10  years 
ago,  you  will  get  the  same  quotient  as  if  you  should  divide 
my  present  age  by  my  age  15  years  ago.  What  is  my  present 
age? 

37.  If  you  divide  my  age  c  years  hence  by  my  age  a  years 
ago,  you  will  get  the  same  quotient  as  if  you  should  divide 
my  present  age  by  my  age  d  years  ago.     What  is  my  present 


age.'' 


38.  Divide  $415  among  A,  B  and  0  so  that  A  shall  have 
$-10  less  than  B,  and  C  $20  more  than  half  as  much  as  A  and 
B  together. 

39.  Divide  %a  among  0,  D  and  E  so  that  C  shall  have  %m 
less  than  D,  and  E  ^n  more  than  one  third  the  share  of  C  and 
D  together. 

40.  A  can  do  a  piece  of  work  in  20  days,  B  in  24  days,  and 
C  in  30  days.     In  what  time  can  they  .ogether  do  the  work? 

41.  A,  B  and  C  can  do  a  piece  of  work  in  4  days,  A  alone 
in  12  days,  and  B  alone  in  10  days.  How  long  would  it  take 
C  to  do  it? 

42.  A,  B  and  C  can  do  a  piece  of  work  in  6  days,  A  alone 
in  9  days,  and  B  alone  in  12  days.  How  long  would  it  take  C 
to  do  it? 

43.  A  can  do  a  piece  of  work  in  a  days,  B  in  b  days,  and 
C  in  c  days.     In  what  time  can  they  together  do  it? 

44.  A  man  is  12  years  older  than  his  wife;  four  years  ago 
8  times  her  age  was  5  times  his.  What  are  their  present 
ages? 


)ur.     In  how 

311  r,  and  in  n 
)ur.     In  how 

,nd  sold  them 
ars  as  he  now 

md  sold  them 
any  dollars  as 

y  age  10  years 

should  divide 

is  my  present 

ny  age  a  years 
should  divide 
is  my  present 

t  A  shall  have 
auch  as  A  and 

shall  have  %)n 
share  of  C  and 

n  24  days,  and 
do  the  work? 
days,  A  alone 

•  would  it  take 

days,  A  alone 
\rould  it  take  C 

in  /;  days,  and 
Ho  it? 
four  years  ago 
their  present 


PIWBLEMS. 


487 


45.   A  man   is  a  years  older  tlian  his  wife;  b  years  ago  c 
times   her   age   was   in   times  his.      What   are   their   present 


ages? 


46.  Divide  $1200  prolit  so  that  A  may  have  one  fourth 
and  SlOO  more,  B  $o'0  less  than  one  tliird,  and  C  §250  more 
than  one  sixth. 

47.  The  interest  on  i*^  of  a  certain  capital  at  5  i»er  cent 
added  to  the  interest  on  the  remainder  at  G  per  cent  is  equal 
to  $1680.     What  is  the  capital? 

48.  A  person,  asking  the  distance  to  a  certain  city,  was 
told  that  after  he  had  gone  one  fourth  the  distance  and  two 
thirds  the  remaining  distance,  he  would  still  have  20  miles  to 
travel.     What  was  the  distance? 

49.  How  far  can  a  person  who  has  5  hours  to  spare  ride 
at  6  miles  per  hour  so  as  to  walk  back  in  time  at  4  miles  per 
hour? 

50.  How  far  can  a  person  who  has  n  hours  to  spare  ride 
at  h  miles  per  hour  so  as  to  walk  back  in  time  at  c  miles  pc-* 
hour? 

51.  A  man  bought  15  horses  for  $16G5,  paying  $120  for 
each  good  horse,  and  $75  each  for  the  poor  ones.  How  many 
of  each  did  he  buy? 

52.  The  difference  of  the  squares  of  two  consecutive  num- 
bers is  15.     What  are  the  numbers? 

53.  The  difference  of  two  numbers  is  2,  and  the  difference 
of  their  squares  is  28.     What  are  the  numbers? 

54.  The  sum  of  two  numbers  is  12  ;  the  square  of  tlie 
greater  is  48  more  than  the  square  of  the  less.  What  are  the 
numbers? 

55.  The  product  of  two  consecutive  numbers  is  4  more 
than  the  square  of  the  less.     What  are  the  numbers? 

56.  Divide  60  into  three  such  parts  that  one  third  of  the 
first,  one  fourth  of  the  second,  and  one  fifth  of  the  third  shall 
be  equal  to  each  other? 

57.  Divide  80  into  four  such  parts  that  if  the  first  be  in- 
creased by  3,  the  second  diminished  by  3,  the  third  multi})]iL'd 
by  3,  the  results  siiall  be  equal. 

58.  The  greater  of  two  numbers  is  4  times  the  less;  if  each 


488 


i<L)fl  'Ll'J  EQ  UA  TIO^S. 


be  increased  by  3,  the  greater  will  be  3  times  the  less.     What 
are  the  numbers? 

59.  A  mun  is  10  years  older  than  his  wife;  in  10  years 
twice  the  sum  of  their  ages  will  be  G  times  her  present  age. 
What  is  the  age  of  each? 

60.  A  man  bougiit  a  certain  number  of  sheep  for  $1500; 
he  reserved  80,  and  sold  the  remainder  for  $960.  How  many 
did  he  buy? 

61.  A  father  aged  48  years  has  a  son  aged  12.  In  how 
many  years  will  the  age  of  the  father  be  three  times  that  of 
the  son? 

62.  A  merchant  has  two  kinds  of  tea;  one  cost  $1.50  a 
pound,  and  the  other  $2.  He  wishes  to  mix  them  so  as  to 
have  50  pounds  worth  $1.80  a  pound.  How  much  of  each 
must  he  use? 

63.  In  a  certain  quantity  of  mortar  the  sand  was  15  pounds 
more  than  |of  the  whole,  the  lime  9  pounds  less  tlian  I  of  the 
whole,  and  the  plaster- of -paris  G  pounds  less  than  |  the  sand. 
What  was  the  amount  of  the  mortar? 

64.  A  laborer  agreed  to  work  50  days  on  the  condition 
that  he  should  receive  $1.50  for  everyday  he  worked,  and  for- 
feit $0.75  for  every  day  he  was  idle.  At  the  end  of  the  time 
he  received  $48.     How  many  days  did  he  work? 

65.  A  grocer  having  GO  pounds  of  coffee  worth  15  cents  a 
pound  mixed  it  with  so  much  coffee  at  18  cents  a  pound 
that  the  mixture  was  worth  IG  cents.     How  much  did  he  use? 

66.  The  interest  on  a  certain  capital  at  5  per  cent  is  $20 
less  than  the  interest  on  $900  more  at  1  per  cent  less.  What 
is  the  capital? 

67.  A  woman  bought  200  apples  at  5  for  3  cents,  and  sold 
part  at  2  for  a  cent,  and  part  at  5  for  4  cents,  thereby  making 
10  cents.     How  many  of  each  kind  did  she  buy? 

68.  A  and  B  play  at  cards.  A  begins  with  $120,  and  B 
with  $180  ;  when  they  stop  playing  B  has  four  times  as  much 
as  A.     How  much  did  B  win? 

69.  From  a  cask  of  wine  one  fourth  leaked  out,  then  20 
gallons  were  drawn,  when  it  was  found  to  be  10  gallons  les-^ 
than  half  full.     How  much  did  it  hold? 


)  less.     What 

;  in   10  years 
•  present  age. 

>ep  for  ^UOO; 
How  many 

12.     In  how 
times  that  of 

3  cost  $1.50  a 

them  so  as  to 

much  of  each 

was  15  pounds 
3  than  I  of  tlic 
lau  ^  the  sand. 

tlie  condition 
jrked,  and  for- 
nd  of  the  time 

• 

orth  15  cents  a 

ents   a  pound 

ich  did  he  use? 

ler  cent  is  $20 

it  less.     Wluil 

cents,  and  sold 
-hereby  making 

I'? 
th  $120,  and  B 

times  as  much 

id  out,  then  20 
10  gallons  Ics-^ 


PROBLEMS. 


480 


70.  An  estate  of  $4G80  is  to  he  divided  among  4  sons  smd 
3  daughters.  Each  son  is  to  receive  $40  more  than  tiie  next 
younp^r;  the  eldest  daughter  is  to  have  $20  less  than  tiio 
eldest  .^on.  and  each  of  her  sisters  $20  less  than  the  next 
older.     What  did  each  child  get? 

71.  A  sum  of  $2880  is  to  be  divided  among  A,  B  and  C. 
Five  times  A's  share  is  to  be  e([ual  to  three  ♦^imes  C's,  and  B  is 
to  have  twice  as  much  as  A  and  0.     What  does  each  receive? 

72.  Six  plasterers,  8  journeymen  and  12  apprentices  re- 
ceive at  the  end  of  a  certain  time  $387.50.  The  plasterers 
receive  $2  a  day,  the  journeymen  $1.25,  and  the  ap])rentices 
75  cents.     How  many  days  did  they  work? 

73.  In  the  above  problem,  what  should  each  class  of  work- 
men receive  if  each  plasterer  worked  3  days  more  than  the 
journeymen,  and  the  apprentices  G  days  less? 

74.  A  man  Avished  to  give  10  cents  eacli  to  some  beggars, 
but  found  he  had  not  enough  of  money  by  14  cents;  he  then 
gave  each  one  8  cents,  and  found  that  he  had  10  cents  re- 
maining.    How  many  beggars  were  there? 

75.  A  post  is  6  feet  more  than  \  in  the  mud,  2  feet  less 
than  \  in  the  water,  and  4  feet  in  the  air.  What  is  the  length 
of  the  pole? 

76.  A  and  B  begin  trade.  A  has  $1000,  and  B  $1210. 
The  former  gains  a  certain  per  cent  on  his  investment,  and 
the  latter  loses  the  same  per  cent,  when  their  capitals  are  found 
to  be  equal.     What  was  the  amoant  lost  and  gained? 

77.  A  person  in  play  lost  \  of  his  money,  then  won  $G0, 
after  which  he  lost  |  of  what  he  then  had,  Avhen  he  found  he 
had  but  $350  remaining.     What  had  lie  at  first? 

78.  In  a  camp  of  3294  soldiers  there  were  3  cavalry  to 
every  2G  infantry,  and  half  as  many  artillery  as  cavalry. 
What  was  the  number  of  each? 

79.  The  right-hand  digit  of  a  certtfin  number  is  2  less  than 
the  second;  and  if  the  number  be  divided  by  the  sum  of  the 
digits,  the  quotient  will  be  7.     What  is  the  numbei'? 

80.  The  length  of  a  town  lot  exceeds  its  width  by  12  feet. 
If  each  were  3  feet  greater,  there  would  be  an  increase  of  G45 
square  feet  in  its  dimensions.     What  is  the  lengtli? 


^-^ 


490 


t^IMPL  M  EQ  UA  TWAS. 


»        1 

f 

I 

»» 


[  i      • 


f       » 


'Vi 


M 


8i.  A  house  was  sold  for  I^OSOO,  by  wliich  tliere  was  a  Cf'- 
tain  gain.  If  it  luul  been  sold  for  I'lOOO  less,  3  times  the 
resulting  loss  would  have  been  twice  the  i)resent  gain.  What 
was  the  cost  of  the  house? 

82.  A  can  do  a  i)iece  of  work  in  12  daj^s,  and  B  in  15. 
After  A  has  worked  4  davs  B  comes  to  help  him.  In  what 
time  can  they  both  linish  it? 

83.  A  tank  has  two  filling  and  one  emptying  pipe.  One 
can  fill  it  in  12  hours,  the  other  in  24  hours  ;  and  the  third 
can  empty  it  in  18  hours.  If  they  are  started  at  the  same 
time,  how  long  will  it  take  to  fill  the  tank? 

84.  In  the  preceding  problem,  suppose  the  third  can 
empty  it  in  8  hours.     How  long  will  it  take  to  fill  it? 

85.  Suppose  it  is  full  already,  and  the  third  can  empty  it 
in  6  hours.     How  long  will  it  take  to  empty  it? 

86.  A  person  travelled  168  miles,  of  which  he  went  3  by 
boat  and  4  by  coach  to  every  G  by  rail,  and  walked  one  thii'd 
as  far  as  he  went  by  boat.  How  many  miles  did  lie  travel  by 
each? 

87.  The  sum  of  two  numbers  is  42.  If  the  less  be  divided 
by  the  greater,  the  quotient  will  be  less  by  ^  than  when  the 
less  is  divided  by  half  the  greater.  AVhat  are  the  num- 
bers? 

88.  A  and  B  are  of  the  same  age.  Three  times  A's  age 
0  years  ago  is  equal  to  twice  B's  age  9  years  hence.  What  is 
the  age? 

89.  In  tossing  pennies,  A  threw  heads  3  times  out  of  5, 
and  B  4  times  out  of  7.  In  all  they  get  41  heads.  How  many 
times  did  they  toss? 

90.  What  two  numbers  are  those  whose  sum  is  13,  and 
whose  product  added  to  the  square  of  the  less  makes  50? 

91.  A  tank  has  five  i)ipes.  No.  1  can  fill  it  in  G  hours,  No. 
2  in  8  hours,  and  No.  3  in  12.  No.  4  can  empty  it  in  9  hours, 
and  No.  5  in  18.  If  they  begin  at  the  same  time,  how  long 
will  it  take  to  fill  the  tank? 

92.  A  starts  from  a  certain  ]dace,  and  travels  at  the  rate 
of  17  miles  in  5  hours.     One  hour  and  53  minutes  after,  B 


rnoniKMs. 


401 


;re  was  a  c^-- 

,  3  times  the 

gain.     Whiit, 

md  B  in  15. 

im.     In  wliiit 

ig  pipe.  Olio 
and  the  third 
d  at  the  same 

:he   third  can 

fill  it? 

d  can  empty  it 

• 

he  went  3  by 
,lked  one  third 
,id  he  travel  by 

less  be  divided 
lan  when  the 
are   the  nuni- 

e  times  A's  age 
nee.     What  is 

times  out  of  5, 
s.     How  uiany 

mm  is  13,  and 
makes  50? 
in  6  hours.  No. 
y  it  in  9  houv^^. 
:imo,  how  long 

Ivels  at  the  rate 
linutcs  after,  B 


starts  at  the  rate  of  10  Tnih\s  in  \  liours.     IIow  far  will  they 
tra\ol  before  B  overtakes  A? 

93.  Two  i)er.sons  start  from  the  same  place  at  the  same 
time,  going  in  the  same  dire';tion.  One  travels  "^l  mik'S 
an  liour  fa.si<>r  than  the  other.  Afu-r  they  had  gone  as  manv 
hours  as  the  slower  goes  miles  per  hour,  their  distance  ajiart 
was  equal  to  half  tiie  distance  travelled  by  the  faster,  liow 
long  did  they  travel? 

94.  Two  men  travel  in  opposite  directions;  the  rate  of 
one  is  1  mile  more  than  two  thirds  the  rate  of  the  other. 
When  they  had  gone  4  hours  the  distance  iipart  was  equal  to 
44  miles.     What  were  their  rates? 

95.  An  officer  in  arranging  his  men  in  the  form  of  a  scjuare 
found  that  he  needed  5  men  to  comjileto  the  square,  and  hy 
increasing  the  file  by  6  and  diminishing  the  rank  by  5  he  had 
5  men  too  many.     How  many  men  had  he? 

96.  A  coach  that  travels  G  miles  an  hour  starts  50  niiiiutes 
after  another  that  goes  5  miles  an  hour.  How  far  will  the 
iirst-named  travel  in  order  to  be  11  miles  ahead  of  the  other? 

97.  A  merchant  withdrew  from  his  caj)ital  8500  at  (he  end 
of  each  year  for  current  expenses;  his  profits  e.ich  year  were 
33^  per  cent  of  his  unexpended  capital.  In  3  years  his 
oi'iginal  stock  was  doubled.     What  was  his  original  stock? 

98.  What  fraction  is  that  whose  denominator  is  2  more 
than  the  numerator,  and  if  3  be  subtracted  from  both  numer- 
utor  and  denominator  the  friiction  Avill  be  |? 

99.  Divide  40  into  two  such  parts  that  the  greatei'  dimin- 
ished by  4  and  divided  by  the  less  increased  by  G  shall 
be  nj 

100.  On  a  note  interest  is  ]iaid  at  0  per  cent.  At  the  end 
of  the  first  year  $200  is  credited  on  the  princi])al,  and  the  lato 
of  interest  is  reduced  to  5  per  cent,  when  the  annual  interest 
is  diminished  by  one  fifth.     What  was  the  face  of  the  note? 

loi.  The  difference  between  the  simple  and  compouinl  in- 
interest  of  a  certain  principal  during  the  second  year  at  5  ]ttr 
cent  is  810.     What  is  the  principal? 

102.  The  fore  and  hind  wheels  of  a  carriage  have  circum- 
ferences of  12  and  IG  feet.      IIow  far  will  the  earriaae  have 


492 


SIMPL  M  h'Q  ITA  TIO NS. 


i    t 

i 

>) 

1 

1 

r 

f  .. 


gone  when  the  sum  of  the  revolutions  made  by  the  wliecls 

is^iS?? 

[03.  During  the  first  year  a  broker  gains  20  per  cent  on 
his  capital,  the  second  year  he  gains  30  per  cent  on  his  in- 
creased capital,  and  the  third  25  per  cent  on  his  re-increased 
capital,  when  he  tinds  tlnit  his  capital  is  $4'J10  more  than 
what  he  began  with.      What  was  his  lirst  cai)iLal? 

104.  A  man  sold  a  house  and  furniture  for  JjiO-lOO;  |  of  the 
price  of  the  house  was  $200  less  than  f  the  i)rice  of  the  furni- 
ture.    What  was  the  value  of  eacii? 

105.  A  purse  contains  05  coins,  part  cents  and  part  dimes. 
How  many  of  each  are  thtm  if  the  total  value  is  ^2? 

106.  Each  member  of  a  base-l)all  club  subscribes  as  many 
cents  as  there  are  members.  If  there  had  been  10  more 
members,  each  subscription  would  have  been  0  cents  less. 
How  many  members  were  there? 

107.  A  man  })urchase(i  a  number  of  lemons  at  2  cents 
each,  and  ^  as  many  at  3  cents  each;  he  sold  them  all  at  the 
rate  of  2  for  5  cents,  and  gained  25  cents.  How  many  of 
each  kind  did  he  i)urchase? 

108.  A  boy  in  Hying  his  kite  lost  -f  of  his  string,  then 
added  65  feet,  and  found  that  it  was  just  |  of  its  original 
length.      What  was  the  length  at  first? 

109.  A  and  B  start  from  two  towns  that  arc  133  miles 
apart  and  travel  towards  each  other.  They  meet  at  the  end 
of  10  hours,  and  find  that  A  has  travelled  1|-  miles  an  hour 
more  than  B.     How  many  miles  had  each  travelled? 

no.  A  man  owning  a  cow  and  horse  found  that  4  loads  of 
liay  would  keep  them  both  6  months.  Having  disposed  of 
his  horse,  he  found  that  the  same  quantity  of  Iwiy  would  last 
the  cow  14  months.     How  long  would  1  load  last  each? 

111.  A  has  $647,  which  is  $33  less  than  4  times  what  I) 
has;  C  is  worth  twice  as  much  as  A  and  B  together,  lacking 
$72.     How  much  have  B  and  C? 

112.  A  boat  which  could  move  14  miles  in  still  water  was 
accelerated  2|  miles  })er  hour  going  down  stream,  and  retarJeil 
the  same  returning;  it  was  IG  hours  longer  coming  up  a  cer- 
tain distance  tliaii  going  down.     What  was  the  distance? 


rnonuiMs. 


4  on 


by  the  wheels 

20  per  cent  on 
cent  on  his  in- 
li.s  re-inci-e:iseil 
•jlO  more  tluiu 

ce  of  the  furni- 

iind  part  dimes, 
is  i^2? 

)seribes  us  many 
I  been  10  more 
311  1)  cents  less. 

Tions  at  2  cents 
I  them  all  at  the 
.     IIow  many  of 

his  strinj;,  then 
V  of  its  original 

at  arc  13;i  miles 
J  meet  at  the  end 

\  miles  an  hour 
iivelled  ? 

nd  that  4  loads  ol" 
iving  disposed  of 
of  Iwiy  -svonld  lasl 
il  last  each? 
u  4  times  what  1> 

toi^ether,  lackini,^ 

in  still  water  was 
•earn,  and  retaracil 
r  coming  np  a  eer- 

the  distance? 


113.  A  and  B  liave  tli^same  income.  A  spends  I  of  liis, 
and  B  by  8i)ending  i&2()0  a  year  more  tiian  A  finds  Iiimsclf  at 
tiic  end  of  0  years  )5^4."i()  in  debt.     Wiiat  was  their  income? 

114.  A  farmer  i)oug]it  22  cows  at  a  certain  ])rice;  liad  ho 
])aid  S  per  (;ent  less  lie  conld  have  pnrciiased  1  more  cow  and 
had  ^21  left.     Winit  was  tlie  [)ri(!e  of  each  cow? 

1 15.  A  son  is  i  the  age  of  iiis  father,  and  1 1  years  ago  lie 
was  I  of  his  age.     How  old  is  each? 

1 16.  A  man  rows  5  miles  an  hour  in  still  water.  How  fai 
can  ho  row  n})  a  stream  and  back  in  3  hours,  the  stream 
ilowing  a  mile  an  hour? 

1 17.  A  man  bought  some  sheep  for  ^94.  Having  lost  7  of 
them,  he  sold  {  of  the  remainder  at  lirst  cost  for»^2(>.  How 
many  did  he  buy? 

118.  The  pe4'imeter  of  a  rectangle  is  28  feet;  if  2  feet  be 
taken  from  its  length  and  added  to  its  breadth,  its  area  is  in- 
creased by  12  square  feet.     Find  its  original  breadth? 

119.  A  man  can  row  9  miles  an  hour  with  the  stream, 
and  3  against  it.  How  far  can  he  go  so  as  to  be  back  in 
(1  hours? 

1 20.  The  first  digit  of  a  certain  number  exceeds  the  second 
by  5,  and  if  the  digits  be  inverted  the  new  number  will  be  % 
of  the  original  number.     What  is  the  number? 

121.  Divide  $900  in  two  such  parts  that  the  interest  on 
one  part  at  4^  per  cent  may  exceed  that  on  the  other  at  3i  per 
cent  by  50  cents. 

122.  How  much  foreign  brandy  at  18  a  gallon  and  whisky 
at  $3  a  gallon  must  be  mixed  together  so  that  the  compound 
may  be  sold  for  $9,  and  the  merchant  thereby  gain  30  i)er  cent. 

123.  A  person  has  two  kinds  of  coins.  Four  pieces  of  one 
make  a  dollar,  or  10  pieces  of  the  other.  How  many  of  eacli 
must  be  taken  so  as  to  have  7  pieces  equala  dollar? 

124.  Find  two  numbers  Avhose  product  is  72,  and  wliosc 
(lifTerence  multiplied  by  the  greater  is  found  by  subtracting 
the  product  from  18  times  the  greater. 

125.  A  person  after  spending  $200  more  than  \  of  his  in- 
come had  remaining  $75  less  than  \  of  it.  What  was  his  in- 
come? 


494 


."^IMl'LE  KqUATlONS. 


( 


> 


i   , 


126.  Divide  77  into  two  siicli  |)!irt.s  tliat  tlio  quotient  of 
the  first  divided  by  8  added  to  tlie  ({uotieut  of  the  second 
divided  by  0  .siiuil  be  0? 

127.  The  sum  of  tliroo  num))ors  is  155.  If  llie  sooond  be 
divided  by  the  (irnt,  tiio  (|U()tient  is  )l,  and  2  I'or  u  reniaiudei". 
tiiul  the  tliird  divided  by  the  secoiul  gives  13  for  u  (|UoLieMt  jin.l 
'A  for  a  reiuuiiider.     Wbiit  are  tlio  numbers? 

uS.  At  a  ball  there  were  twiee  as  many  gentlenuMi  as 
ladies.  When  8  cou[)les  danced  tiiere  were  remaining  three 
times  as  many  gentlemen  as  hidies.  What  was  the  number  of 
each? 

129.  A  can  build  7  cubic  yards  of  wall  in  4  days,  li  12 
yards  in  5  days,  and  C  9  yards  in  2  days.  How  long  will  it 
take  all  three  to  build  850  yards? 

130.  Each  of  the  Hiree  digits  of  a  certain  number  is  greater 
than  the  next  following  by  1;  when  the  digits  are  inverted, 
the  new  number  will  be  18  more  than  ^  the  first  number.  What 
is  the  number? 

131.  A  farmer  bought  30  sheep  and  10  calves  for  the  same 
sum.  If  the  sheep  had  cost  25  per  cent  more  and  the  calves 
35  ])er  cent  less,  7  sheep  would  have  cost  $;3  more  than  4 
calves.     What  did  each  sheep  cost? 

132.  Ui)on  withdrawing  from  the  business  A  takes  ^  of 
the  capital  and  $100  more,  B  ^  of  the  new  remainder  and 
<;100  more;  C  gets  $300.     What  was  the  ca])ital? 

133.  What  number  i-  that  which  gives  the  same  continued 
product  when  divided  i'lco  3  ecpuil  parts  as  when  divided  into 
4  equal  ])arts? 

134.  Find  a  number  of  two  digits,  the  first  of  which  is  4 
times  the  second,  and  the  number  is  2  less  than  3  times  tlu' 
number  formed  by  inverting  tlie  digits. 

135.  In  going  fi'om  one  town  to  another  a  traveller  found 
at  a  certain  })lace  that  the  distance  travelled  was  ^  the  wliolo 
distance,  and  when  he  had  gone  11  miles  further  he  had  I  of 
the  whole  distance  yet  to  go.     What  was  the  distance? 

136.  A  wine-merchant  has  wine  in  casks  of  two  sizes.  One 
containing  2|  gallons   he  charges  $8.50  for;    the  other,  .'J| 


PRODLKMS. 


40.1 


[)  quotient  of 
if  tlio  second 

,liu  second  he 

a  roniainder. 

I  (juutient  iui>l 


rrcntlenien  jis 
iujiininf^  three 
the  niuuhcr  of 


k  4  days,  B  12 
o\v  long  will  it 

mher  is  greater 
,s  are  inverted, 
number.  AVhut 

res  for  the  same 
3  and  the  calves 
,a  more  than  4 

3s  A  takes  \  of 
remainder  and 

il? 

same  coutiniicd 

len  divided  into 


^t  of  which  is  4 
,au  3  times  tlu' 


I  traveller  fouiul 
was  4  the  whole 
ther  he  had  I  of 
listaiice? 
f  two  sizes.  One 
••    the  other,  ;5.i 


gallons,  la  priced  at  1<1(>.00.     What  is  tlio  price  of  (lie  ca.sks, 
supposing  Lliem  to  cost  the  sanu'? 

137.  A  nmn's  income  was  >i>%\\()  tlie  first  year,  :ui  1  increased 
$50  each  sucuteding  year.  At  the  end  of  :)  ycar.s  he  had 
Liived  «»ir).T5.      What  were  his  annual  expenses? 

138.  If  A  ^ives  B  *1(>  he  will  liave  twice  as  much  as  B; 
but  if  13  i;ives  A  ^BlU  he  will  have  ^  as  nuich  as  A.  llow  miicli 
had  UMiiii 

(M«o.) 


PriOTiLEMS       INVOLVING       EQUATIONS       WITH 
MOKE     IJNKNOVVN     WUANIMllia. 


TWO      OH 


1.  It  is  found  that  when  a  ship  steams  VI  knots  (sea-miles) 
an  hour  witli  the  waves  she  pitches  1  in  15  seconds,  and 
steaming  at  the  same  si)eed  against  them  she  ])iti'hes  1  in  G 
seconds.  What  is  the  speed  of  the  waves,  and  how  many 
w:ives  arc  there  in  a  sea-mile? 

2.  'Vwo  men  start  at  the  same  time  to  make  the  same 
journey.  The  tlrst  goes  10  miles  the  first  day,  and  goes  a  cer- 
tain fixed  distance  more  every  following  day  than  he  d.d  the 
(lay  before.  He  overtakes  the  second  at  the  end  of  the  8th 
day,  and  finishes  his  journey  at  the  end  of  the  1 1th,  while  the 
second  finished  at  the  end  of  the  Vli\\.  What  is  the  length 
of  the  journey,  and  how  far  did  the  second  go  each  day? 

3.  A  cannon  being  fired  while  a  heavy  wind  was  blowing, 
it  was  found  that  the  sound  required  4|-  seconds  to  go  a  miio 
with  t)ie  wind,  and  4|-  seconds  to  go  a  mile  against  the  wind. 
What  was  the  velocity  of  the  wind,  and  what  time  would  have 
i)L'en  required  for  the  sound  to  go  a  mile  in  still  air? 

4.  The  greatest  distance  between  Venus  and  the  earth  i:j 
I'lO  millions  of  miles;  the  least,  22  millions.  What  is  tlu!  dis- 
tance of  each  from  the  sun,  supposing  tlu.'t  each  moves  around 
the  sun  in  a  circular  orbit  having  the  sun  in  its  centre? 

5.  A  ])rother  and  sister  being  .isked  how  large  the  family 
was,  the  brother  replied,  "I  have  as  many  brothers  as 
sisters."  The  sister  rei)lied,  ''  I  have  twice  as  many  l)rothers 
us  sisters."     llow  many  boys  and  girls  were  in  the  family? 


.11 


mmmm 


496 


SIMPLE  EqUATIONS. 


I 

N' 


f  i, 


6.  Find  tliat  fraction  whose  value  becomes  ^  wlien  n  is  sub- 
tracted from  ciicii  of  its  terms,  and  I  when  m  is  added  to  each 
of  its  terms. 

7.  Find  two  numoers  such  that  tlieir  difference  is  153, 
and  the  lesser  goes  into  the  greater  9  times  and  1  over. 

8.  One  number  divided  by  another  gives  the  quotient  4, 
with  3  as  a  remainder.  Increasing  divisor  and  dividend  by 
10,  the  quotient  is  )l  and  the  remainder  23.   Find  the  numbers. 

9.  Find  two  quantities  such  that  half  their  sum  added  to 
half  their  difference  shall  make  a,  and  half  their  difference 
subtracted  from  half  their  sum  shall  leave  the  remainder  b. 

10.  Find  two  quantities  whose  sum  and  quotient  are  each 
equal  to  m. 

11.  Find  two  numbers  of  three  digits  of  Avhicli  one  is 
formed  by  simply  reversing  the  order  of  digits  in  the  other, 
and  wJiich  fulfil  the  following  conditions:  (1)  the  sum  of  the 
digits  in  each  is  15;  (2)  the  sum  of  the  first  and  last  digits  is 
3  greater  than  the  second  one;  (3)  the  difference  of  the  num- 
bers is  99. 

12.  Each  of  two  vessels,  A  and  B,  was  partly  filled  with 
water.  A  man  poured  from  A  into  B  as  much  as  was  already 
in  V>,  then  from  B  into  A  as  much  as  was  left  in  A,  then  from 
A  into  B  as  much  as  was  left  in  B,  when  each  vessel  con- 
tained 8  quarts  of  water.    How  much  did  each  contain  at  first? 

13.  Find  two  quantities  the  sum  of  whose  reciprocals  is  5, 
and  4  tlk^  one  added  to  ^  the  other  is  equal  to  twice  their 
product.* 

14.  For  $G.60  one  can  buy  either  20  pounds  of  coffee  and 
25  of  sugar  or  14  of  coffee  and  34  of  sugar.  What  is  the 
price  of  each  per  pound? 

15.  A  river  steamer  can  run  90  miles  down  stream  '.mkI 
back  again  in  15  hours;  but  if  she  runs  120  miles  down, 
she  can  only  get  back  70  miles  on  her  return  journey  at  the 
end.  of  15  hours.  What  is  her  speed  and  the  flow  of  the 
river?  * 

16.  Cn  a  river  were  two  steamers,  the  speed  of  the  swift 

*  Compare  witli  Exercises  11  to  20,  §  138. 


PROBLEMS 


497 


wlien  n  is  sub- 
;  added  to  each 

Perencc  is  153, 
I  1  over, 
the  quotient  4, 
id  diyideiid  by 
id  the  numbers, 
r  sum  added  to 
their  difference 
1  remainder  b. 
lotient  are  each 

)f  which  one  is 
ts  in  the  other, 
)  the  sum  of  the 
;md  last  digits  is 
mce  of  the  num- 

)artly  filled  with 
h  as  was  already 
in  A,  then  from 
each  vessel  con- 
contain  at  first? 
reciprocals  is  5, 
lal  to  twice  their 

nds  of  coffee  and 
ir.     What  is  the 

own  stream  :ni(l 
120  miles  down. 
n  iournev  Jit  the 
the  fiow  of  the 

-)ecd  of  the  swift 


sl38. 


one  being  3  miles  an  hour  greater  than  that  of  the  slow  one. 
A  man  who  went  58  miles  down  the  river  on  the  slow  boat 
and  30  miles  back  on  tiie  swift  one  found  that  he  had  been 
9  hours  on  the  water.  IJut  when  he  went  87  miles  down  on 
the  slow  boat  and  90  miles  back  on  the  swift  one,  he  found 
that  it  took  18  hours.  What  was  the  speed  of  each  boat  and 
the  flow  of  the  river? 

17.  A  quadrilateral  has  four  sides,  a,  h,  c  and  d.  If  }  of 
a  be  added  to  b,  then  \  of  the  extended  b  be  added  to  c,  and 
then  |-  of  the  extended  c  to  d,  the  four  sides  will  each  be 
equal  to  m.     What  was  the  length  of  each  sidr  at  first? 

18.  Three  pedestrians  started  on  a  journey.  The  first 
performed  it  in  a  certr'ii  time;  the  second,  going  1  mile  an 
hour  slower,  took  l;i  hours  ^onger;  the  third,  going  2  miles 
an  hour  slower  than  the  first,  took  33  ho\u6  longer.  What 
was  the  distance,  and  the  speed  of  each? 

19.  The  perimeter  of  a  triangle  whose  sides  are  a,  b,  c,  is 
m  feet.  If  |  the  side  a  be  added  to  b,  then  |  of  the  prolonged 
b  be  added  to  c,  and  then  \  of  the  prolonged  c  be  added  to  a, 
the  sides  will  be  equal.     What  is  the  length  of  each  side? 

20.  Divide  232  into  three  parts,  A,  B  and  C,  sucli  that, 
whether  we  subtract  A  from  the  sum  of  B  and  C,  B  from  \ 
the  sum  of  A  and  C,  or  C  from  '  the  sum  of  A  and  B,  the 
remainders  shall  all  be  equal. 

21.  Find  two  quantities  whose  difference  and  product  are 
each  equal  to  n. 

22.  The  quotient  of  two  niinil)ers  is  2,  and  2  times  their 
sum  is  equal  to  6  times  their  difference.  What  are  the  num- 
bers? 

23.  A  man  has  a  saddle,  worth  850,  and  two  horses.  If  tlio 
c;.udie  be  put  on  horse  A,  he  will  equal  B  in  value;  but  if  })ut 
on  B,  his  value  will  be  double  that  of  A.  What  is  tho  value 
of  each  horse? 

24.  What  number  of  two  digits  is  equal  to  4  times  their 
sum  and  12  times  their  difference? 

25.  What  number  of  two  digits  is  equal  to  4  times  their 
sum,  and  when  the  digits  are  reversed  equal  to  7  times  their 
K'lim? 


498 


SIM  PL  E  EQ  UA  TIONS. 


J 


II' 


L 


26.  Fiiul  a  number  of  two  digits  tli.it  is  equal  to  4  times 
the  sum  of  its  digits  increased  by  3,  and  if  9  be  added  to  the 
iiiiniber  tlie  digits  will  be  reversed. 

27.  Find  a  number  wliicli  is  greater  by  3  than  G  times  the 
sum  of  its  digits,  and  if  9  ije  subtracted  from  the  number  the 
digits  will  be  reversed. 

28.  What  number  is  that  which  is  4  times  the  sum  of  its 
digits,  and  is  3  greater  than  11  times  their  difference? 

29.  What  fraction  is  thtit  wiiich  becomes  -^  whr,n  2  is  added 
to  the  denominator,  and  J^  if  5  be  subtracted  from  the  numer- 
ator? 

30.  Two  drovers  went  to  market  with  shee]>.  A  sold  90 
and  then  had  left  J-  as  many  as  B.  Then  B  sold  72,  and  had 
f  as  many  as  A  reuiaining.     How  many  did  eacli  have? 

31.  A  woman  bought  GO  apples  for  a  dollar,  giving  3  cents 
for  every  2  bad  ones  and  2  cents  each  for  the  good  ones.  Ilow 
many  of  each  did  she  buy? 

32.  Find  a  fraction  that  becomes  J-  when  4  is  added  to  its 
denominator,  or  2  subtracted  from  its  numerator. 

2,2,-  A  marketman  had  4  more  ducks  than  chickens.  Tie 
sold  the  chickens  for  30  cents  apiece  and  the  ducks  for  40 
cents  apiece,  gaining  40  cents  more  than  if  the  prices  had 
been  reversed.     How  many  of  each  had  he? 

34.  A  boy  bought  a  number  of  apples  at  2  cents  each  aiul 
peaches  at  3  cents  each,  paying  $4.36  for  the  whole;  12  of  the 
api)les  were  bad  and  9  peaches  were  rotten.  Ho  sold  the  good 
a[)])les  at  2  for  5  cents  and  the  peaches  3  for  10  cents,  receiv- 
ing 14.50  for  the  whole.     How  many  of  each  fruit  did  he  buy? 

35.  When  I  was  married  I  was  I  older  than  my  wife;  10 
years  after  her  age  was  |  of  mine.  What  were  o\ir  ages  when 
we  were  married? 

T,6.  A  and  B  can  do  a  piece  of  work  in  12  days;  but  if  A 
worked  twice  as  fast  they  could  do  it  in  84  days.  In  what 
time  could  each  of  them  do  it  singly? 

37.  B  and  0  can  do  a  piece  of  work  in  12  days;  with  the 
assistance  of  A  they  can  do  it  in  9  days.  In  what  time  can  A 
do  it  alone? 

38.  A  farmer  sold  GO   fowls,   a  part  turkeys  and  a  part 


PROnLh\]fS. 


499 


^nal  to  4  times 
)e  added  to  the 

lan  0  times  the 
;hc  number  the 

the  sum  of  its 
Eerence? 
when  2  is  added 
rom  the  numev- 

pep.     A  sold  00 
old  T'l,  Jind  had 
aeii  have? 
[',  jriving  3  eents 
Dod  ones.     How 


4  is  added  to  il.-^; 

ator. 

n  chickens.     Tie 

lie  ducks  for  40 

f  the  prices  had 

3  cents  each  and 

whole;  12  of  the 

He  sold  the  good 

10  cents,  receiv- 

fruitdid  he  buy? 

lan  my  wife;  !<> 

•e  our  ages  when 

2  days;  but  if  A 
^  days.     In  ^vhat 

2  days;  with  tlic 
what  time  can  A 

•keys  and  a  part 


chickens;  for  tkc  turkeys  he  received  ^1.10  apiece,  and  for 
tlie  cliickens  50  cents  apiece,  receiving  for  the  whole  st^-ll.r.O. 
How  many  were  there  of  eacii? 

39.  A  tank  has  4  pipes.  A,  15,  C  and  D.  A,  ]>  and  C  ca-i 
fill  it  in  <)  hours;  B,  C  and  D,  in  8  hours;  C,  D  and  A,  in  10 
hours;  1),  A  and  B,  in  12  hours.  How  long  will  it  take  each 
and  all  to  lill  it?     Ex})lain  the  negative  result  for  I). 

40.  A  tank  has  two  pijies,  of  which  one  may  be  made  to 
run  either  in  or  out.  If  both  run  in  the  tank  is  filled  in  2 
hours;  if  one  in  and  the  other  out,  in  5  iiours.  In  what 
times  would  the  separate  pipes  fill  it? 

41.  A  grocer  bought  50  pounds  of  sugar  and  100  pounds 
of  coffee  for  I2(!.  He  sold  the  sugar  at  an  advance  of  :25  per 
cent  and  the  cofi'ee  at  a  discount  of  10  per  cent,  receiving 
$25.50  for  the  whole.  What  was  the  buying  and  selling  price 
of  each  ? 

42.  Find  the  sum  of  two  numbers  the  difference  of  whose 
squares  is  equal  to  the  difference  of  the  numbers. 

43.  Divide  168  into  three  such  })arts  that  the  second  divi- 
ded by  the  first  gives  5  as  a  quotient  and  10  for  a  remtiinder, 
and  the  difference  between  tlie  third  and  second  muliiplic'd 
by  3  is  equal  to  4  times  the  first. 

44.  A  father  is  5  times  as  eld  as  his  son.  Six  years  hence 
he  will  be  only  3  times  as  old.     AVhat  are  their  present  ages? 

45.  The  sum  of  the  ages  of  two  persons  is  §  of  what  it 
will  be  12  yejirs  hence.  The  dilTerence  between  their  ages 
is   ^   of  what   it  will    be  24  years  hence.     What  are   their 


iiges; 


46.  A  farmer  sold  to  one  person  40  bushels  of  oats  and  30 
})ushels  of  wheat  for  ^44.50,  and  to  another  the  snme  atnouiit 
of  oats,  at  10  cents  a  bush'^]  more,  and  wheat,  at  5  cents  a 
bushel  less,  for  $57.     What  was  the  price  per  bushel  of  each? 

47.  There  is  a  number  of  3  digits  whose  sum  is  10.  The 
first  and  second  is  4  times  the  third,  and  if  207  be  added  tlio 

be  reversed.      What  is  the  innnber? 

48.  There  is  a  numljcr  of  3  digits  wliose  fii'st  and  third 
digits  aie  6  more  than  the  second.  Four  times  tlie  lii'st  is 
11  more  than  the  dilfeience  between  the  second  and  third;  and 


digits  wi 


u% 


I   f 


'l     H 


500 


-67 J//  'L  i:  KQ  I' A  Tl  O^S. 


if  97  be  udded  to  the   nunilier  the  digits  will  be  reversed. 
AVhat  is  the  luiniberi' 

49.  A  certuiu  minibor  of  3  digits  is  34  times  the  sum  of 
its  digits,  Hiid  also  102  times  the  dilference  between  the  first 
and  seeond;  and  if  30  be  iidded  to  the  number  the  second  luul 
third  will  exchange  places.     Wliat  is  the  number? 

50.  An  oarsman  who  can  row  ^0  miles  and  back  in  T 
hours  linds  that  he  can  row  10  miles  with  ihe  current  iu  ilic 
same  time  that  it  takes  him  to  go  4  miles  in  the  contrary 
direction.     Find  the  rate  of  the  current. 

51.  A  merchant  has  two  kinds  of  sugar;  one  cost  8  cents  a 
jiound,  and  the  other  11  cents.  How  much  of  each  must  be 
taken  to  make  120  pounds  woj'th  9  cents  per  pound? 

52.  A  grocer  mixed  tea  that  cost  $1.10  a  i)ound  with  tea 
that  cost  95  cents  per  pound.  The  cost  of  the  mixture  is 
$101.  He  sells  it  at  $1  a  pound  and  gains  $2.  How  many 
pounds  of  each  did  he  use? 

53.  A,  B  and  0  can  earn  $25  in  5  days;  B  and  C,  $28  in  T 
days;  A  and  C,  $22  in  8  days.  What  does  each  man  earn  in  1 
day? 

54  A  and  B  can  do  a  i)iece  of  work  in  2  days;  A  and  C,  4 
times  as  much  in  9  days;  A,  B  and  C,  11  times  as  much  in  is 
days.     In  how  many  days  could  each  do  it  alone? 

55.  A  sum  of  money  at  simple  interest  amounted  in  5  years 
to  $1500,  and  in  8  years  to  $1080.  What  was  the  principal 
and  rate? 

56.  A  person  has  $1200  invested  at  a  certain  rate  and  for 
a  certain  time;  had  the  rate  been  1  per  cent  less  and  the  time 
2  years  more,  he  would  have  had  $24  more  interest;  while 
with  a  rate  2  per  cent  less  and  a  time  1  year  more  he  would 
have  had  $144  less  interest.     Find  the  rate  and  time. 

57.  A  sum  of  money  at  simple  interest  for  c  years 
amounted  to  t  dollars,  and  the  same  for  b  3'ears  amounted  l<> 
i?.  dollars.     W^hat  was  the  princi]>al  and  rate? 

58.  In  a  race  over  a  course  4000  feet  long  A  gives  B  300 
feet  start,  and  wins  by  1  minute  and  20  seconds.  In  a  seconl 
trial  A  gives  him  40  seconds  start,  and  Avins  by  900  feet. 
What  was  the  rate  of  each? 


rnoBLEMti. 


501 


be  reversed. 

es  the  sum  of 

ween  the  first 

he  second  and 

jr? 

nd  hack   in  7 

current  in  Liu- 

L   the  contrary 

3  cost  8  cents  a 
each  must  he 
ound? 

pound  with  tea 
the  mixture  is 
2.     How  many 

and  C,  $28  in  7 
li  man  earn  in  1 

avs;  A  and  C,  4 
s  as  much  in  1^ 

)ne? 

unted  in  5  years 

as  the  principal 

lain  rate  and  fcr 
;ss  and  the  tiim' 
interest;  while 
more  he  wouUl 
id  time, 
lest  for  c  years 
lirs  amounted  l" 

g  A  gives  B  300 
ds.  In  a  secoivl 
Ins   by  900   feet. 


59.  A,  B  and  C  promised  to  give  $1000  to  a  chuvch.  A  gave 
one  liiird  less  tliau  he  agreed  to,  so  B  increased  liis  by  one 
fourth,  wliich  left  855  more  for  C.  Now  if  B  had  given  one 
fifth  less  than  ])roniised,  and  C  $T0  more,  A's  share  would 
have  been  his  original  subscription.  What  was  the  amount 
of  the  first  pledge? 

60.  The  fore  wheels  of  a  carriage  are  10|  feet  in  circum- 
fcrciiue,  and  the  hind  wiieels  13.  In  going  a  journey  the 
fore  wheels  nnikc  ;25O0  more  revolutions  than  the  hind  wheels. 
What  was  the  distance? 

61.  A  coach  has  2  more  outside  passengers  than  inside. 
Six  outsiders  could  travel  at  an  expense  of  $1  more  than  -i  in- 
siders. The  fare  of  all  amounted  to  $30.50.  At,  the  end  of 
half  the  journey  2  were  added  to  the  outside  and  1  inside, 
Avhich  increased  the  total  fare  by  $2.50.  What  was  the  num- 
ber and  fare  of  each  class? 

62.  A  person  has  two  creditors;  at  one  time  he  pays  chem 
$680,  giving  to  one  f  of  the  sum  due  him.  and  to  the  other 
$40  more  than  \  of  his  debt;  at  another  time  he  pays  them 
$580,  giving  to  the  first  f  of  what  remains  due  to  him,  and  to 
the  other  ^  of  what  remains  due  to  him.  What  was  the 
amount  of  each  debt? 

63.  If  a  certain  croquet-ground  were  5  feet  longer  a'^]  3 
feet  broader  it  would  contain  320  more  feet;  but  if  it  were  3 
feet  longer  and  5  feet  broader  it;  would  contain  310  more  feet. 
What  is  its  present  area? 

64.  The  suni  of  two  numbers  is  12,  and  the  difference  of 
their  squares  is  24.     What  are  the  numbers? 

65.  Two  boats,  320  and  300  feet  long  respectively,  are 
moving  with  uniform  speed.  If  they  go  in  opposite  directions 
it  requires  10  seconds  to  pass  each  other;  but  if  they  go  in  the 
same  direction  it  takes  90  seconds  for  them  to  pass.  What  is 
the  speed  of  each  boat? 

66.  A  train  runs  a  certain  distance  at  a  uniform  rate.  If  the 
rate  be  increased  bv  5  miles  an  hour  the  distance  would  be 
travelled  in  f  of  the  time;  but  if  the  rate  be  diminished  by  5 
miles  an  hour  the  time  would  be  increased  by  3  hours.  What 
is  the  rate  and  distance? 


603 


SIMPL IC  KQ  UA  TIONS. 


67.  Wliiit  number  of  3  digits  is  gro'^.tcr  by  09  when  its 
digits  are  reversed;  greater  l)y  270  tliaii  the  sum  of  its  digits; 
iuid  gresiter  by  45  thun  when  the  second  jind  third  ure  truiis- 
posed  ? 

68.  A  ;ind  I5  could  luive  completed  u  certain  ])iece  of  work 
ill  I'-i  duys;  but  ul'ter  both  h;id  worked  4  d;iys  B  was  left  to 
linisli  it  alone,  which  he  did  in  'l\  days  more.  How  long 
would  it  have  taken  each  to  do  it  alone? 

60.  A  number  consists  of  il  digits  whose  sum  is  13,  and  if  15 
be  subtracted  from  the  uunii)er,  and  the  remainder  bedivideil 
by  3,  the  digits  w'll  be  inverted.     What  is  the  number? 

70.  A  boy  spent  his  money  in  oranges.  If  he  had  bought 
5  more,  each  orange  wouhl  have  cost  a  half-cent  less;  if  3  less, 
a  half-cent  more.  How  much  did  he  spend,  and  how  numy 
did  he  l)uy? 

71.  A  person  bought  apples  at  4  cents  a  dozen,  and  U 
times  as  many  peaches  at  12  cents  a  dozen;  after  mixing  them 
ho  sold  them  at  8  cents  a  dozen,  losing  4  cents  on  the  whole. 
How  many  dozen  of  tj'ch  did  he  buy? 

72.  Fi«nd  a  fraction  that  becomes  |  when  2  is  added  to 
its  numerator,  and  \  Avhen  4  is  added  to  the  denominator. 

73.  Five  pounds  of  tea  and  12  ])ouiu]s  of  sugar  cost  $7.44. 
If  tea  Avere  to  rise  10  per  uent  and  sugar  fall  25  j)e:  cent,  8 
])ounds  of  tea  and  0  pounds  of  sugar  would  cost  $11.10.  What 
is  the  price  per  })ound  of  each? 

74.  A's  income  is  half  as  much  again  as  B's,  while  his  ex- 
penses are  twice  as  great  as  B's.  A  spends  $60  more  than  his 
income,  and  B  $00  less  than  his.   What  is  the  income  of  eaeli? 

75.  A  invested  some  money  at  5  ])er  cent,  and  B  at  G  per 
cent,  both  receiving  the  same  amount  of  income.  If  A  had 
invested  $1000  more  than  he  did,  his  income  would  have  been 
11  per  cent  on  B's  investment.     What  did  each  invest? 

76.  An  oarsman  can  row  9  miles  up  stream  and  13  miles 
down  in  4  hours,  or  13  miles  up  and  9  miles  down  in  5  hours. 
What  is  the  rate  of  the  stream  and  of  the  rowing? 

77.  Six  years  lience  the  ju-oduct  of  two  people's  ages  will 
be  greater  by  348  thjm  it  is  now.  What  will  then  be  the  sum 
of  their  ages? 


PliOBLEMS. 


,503 


y  09  when  its 
11  of  its  digits; 
lird  iiro  tnuis- 

])iece  of  work 
;  B  was  left  to 


e. 


How  long 


is  12,  and  if  15 
ulcr  be  divided 
number? 
he  had  bought 
t  less;  if  3  le><s, 
and  how  many 

dozen,  and  1  i 
jr  mixing  them 
s  on  the  whole. 

.  2  is  added  to 
enominator. 
igar  cost  $7.44. 
25  pel   cent,  8 
,t  $11.10.  What 

's,  Avhile  his  ex- 
0  more  than  his 
income  of  each? 

and  B  at  G  per 
ome.     If  A  had 
ivould  have  been 
jh  invest? 
im  and  13  miles 

own  in  5  hours. 

ing? 

eople's  ages  will 

then  be  the  sum 


78.  A  invests  money  at  4  i)cr  cent,  B  at  5  ])er  cent,  and  (J 
at  6  per  cent.  A  and  IJ  together  receive  $5(;o,  B  and  C  $52(», 
and  A  and  C  SOOO.     How  much  does  each  invest? 

79.  Find  the  fpiotient  of  two  numbers  wliuse  sum  is  ;/. 
times  their  dillerence. 

80.  A  and  B  can  tinish  a  job  in  12  days.  A  worked  2  days, 
and  B  3.  How  long  will  it  take  C  to  liiii.sh  it  if  he  could  have 
done  the  whole  in  15  days  with  B's  assistance,  and  iu  10  davs 
with  A's? 

81.  A  carpenter  and  apjU'cntice  received  $10.^0  for  7  days' 
wages,  the  carpenter  getting  20  cents  more  for  2  days'  woik 
than  the  boy  for  3  days'.     What  was  the  daily  wages  of  eacji? 

82.  A  man  paid  $50  for  7  photograi)hs  aiul  12  i)rints;  if 
he  had  paid  $1  more  he  could  have  had  7  })rints  and  15 
photographs.     What  was  the  i)rice  of  each? 


Ratio  jiiid   Proportion. 

i?  104. 

1.  Divide  120  into  three  parts  that  shall  1)0  projiortioual 
to  the  numbers  3,  4,  7. 

2.  Find  two  fractions  that  shall  be  to  each  other  ;is  3:1. 
and  whose  sum  shall  be  %. 

3.  Divide  .0444  into  three  })arts  that  shall  be  to  ea<  h  other 
as  -^  :  i  :  |. 

4.  Find  two  numbers  which  are  to  each  other  as  4  :  3,  and 
whose  difference  is  ^  of  the  less  ? 

5  If  a-  :  //  ::  6  :  8  and  4.^;  —  3_?/  =  7.  what  is  the  value  of 
X  and  //? 

6.  A  year's  profits  were  divided  among  two  ])artncrs  in  the 
proportion  of  3:4.  If  the  second  should  gi'  l'  $425  to  the 
first,  their  shares  would  be  Cfpud.  AVliat  was  the  amount 
divided? 

7.  In  a  first  year's  partnership  A  hnd  3  .-Imres.  and  I>  4. 
In  the  second,  A  had  I,  and  B  2.  In  the  first  year  A  gained 
$300  more  than  he  did  the  second,  and  B  gained  $200  less 
than  he  did  the  second.     What  were  the  i)rofiis  each  vear  ? 


504 


RATIO  AND  PROPORTION. 


;*'». 


'  ( 


t      t 


c, 


c. 


8.  111  .1  fiirni-yard  there  are  4  sheep  to  every  3  cattle,  Jind 
5  cuttle  to  (j  hogs.  How  many  hogs  are  tlu'i-e  to  every  :^0 
sheep? 

9.  A  drover  started  to  market  with  a  herd  of  7  horses  to 
every  5  mules,  lie  sold  'Z'7  horses  and  bouglit  13  mules,  and 
then  had  13  horses  to  every  4  mules.  J  low  many  of  each  had 
he  at  first? 

10.  Find  two  quantities  whose  sum,  difference  and  product 
are  i)roportional  to  5,  1  and  12. 

11.  Wh.it  number  is  that  to  which  if  2,  6  and  12  be  sever- 
ally added,  the  first  sum  shall  be  to  the  second  as  the  second 
is  to  the  third? 

12.  What  two  numbers  are  to  each  other  as  3  to  4,  and  if 
4  ])e  added  to  each  the  sums  will  be  as  4  to  5? 

13.  Wiiat  quantity  must  be  taken  from  each  term  of  the 
ratio  m  :  n  that  it  may  C([nal  the  ratio  c  :  d? 

14.  U  a  :  I)  be  the  S(juare  of  the  ratio  of  r^r  -|-  <^  •  ^ 
show  that  c  is  a  mean  proportional  between  a  and  I). 

15.  If  n  :  b  =  b  :  c,  show  that  a  :  a  -\-  b  —  a  —  b  \  a 

16.  And  under  the  same  conditions  show  that 

{a'  +  h')  {b'  +  c')  =  {ab  +  bc)\ 

17.  li  a  \  b  =  c  \  d,  show  that 

a  {a -^  b  -\-  c  -\-  d)  =  {a -{- b)  {a  ■\-  c). 

18.  In  a  milk-can,  the  quantity  of  milk  is  to  the  entire 
contents  (milk  and  water)  as  5  :  6.  Five  gallons  are  sold,  and 
1  gallon  of  water  is  added;  then  the  ratio  of  the  milk  to  the 
whole  is  4  :  5.     How  many  gallons  of  each  were  there  at  first? 

19.  In  a  two-mile  race  between  a  bicycle  and  a  horse,  their 
rates  were  as  5  to  6.  The  bicycle  had  1  minute  start,  but 
was  beaten  by  312  yards.     What  was  the  rate  of  each? 

20.  A  line  is  divided  by  one  point  into  two  parts  in  the 
ratio  of  3  :  5,  and  by  another  point  into  two  parts  in  the  ratio 
of  1  :  3.  The  distance  between  the  points  of  division  is  1 
inch.     What  is  the  length  of  the  line? 

21.  The  sum  of  the  two  digits  of  a  number  is  6,  and  the 
numbar  is  to  the  number  expressed  by  the  same  digits  reversed 
as  4  :  7.     What  is  the  number? 

22.  One  ingot  contains  two  parts  of  gold  and  one  of  silver, 


PllOliLKMS. 


CO;') 


3  cattle,  and 
to  every  '^0 

•  7  liorses  to 
3  mules,  and 
■  of  each  bad 

3  and  product 

\Vl  be  sever- 
as  the  second 

3  to  4,  and  if 

I  term  ol  the 

%  ^  c  \  b  -r  c, 

lid  b. 

a  —  b  :  a  —  c. 

at 


c). 

to  the  entire 
are  sold,  and 
e  milk  to  the 
there  at  lirst? 
a  horse,  their 
ulc  start,  but 

)f  each? 

0  parts  in  the 
ts  in  the  ratio 

i  division  is  1 

r  is  6,  and  the 
digits  reversed 

d  one  of  silver, 


I 


and  another  two  parts  of  gold  and  three  of  silver.     If  e({ual 
parts  are  taken  from  each  ingot,  what  will  be  the  proportion 

of  the  gold  to  the  silver  in  the  alloy? 

23.  If  two  ounces  be  taken  from  the  first  and  t...ce  from 
the  second,  what  will  be  the  ratio  of  the  gold  to  the  silver? 

24.  A  cask  contains  \  gallons  of  water  and  18  g;dh)ns  of 
alcohol.  How  many  gallons  of  a  mixture  containing  'I  parts 
svater  and  5  parts  alcohol  must  be  put  in  the  cask  so  that  thei"e 
may  be  2  ])arts  of  water  to  7  of  alcohol? 

25.  Which  is  the  greater  I'atio,  I  -\-  a  :  \  —  a  or  I  -\-  a''  : 
1  —  (1.^,  a  being  positive  and  less  than  1? 

26.  Which  is  the  greater  ratio,  a"^—  ah  -f  //  :  c/"  -|-  ab  -f-  // 
or  «^  —  d'W  +  b^  '  «*  +  c^b"^  +  ^\  f^  ii"d  b  having  like  signs? 

27.  What  number  must  ^  taken  from  the  second  U-rm  <.r 
the  ratio  2  :  34  and  added       the  first  that  it  may  ('(pial  5  :  G? 

28.  What  number  must  be  taken  from  each  term  of  the 
ratio  19  :  30  that  it  may  equal  the  ratio  1:2? 

29.  11  (f  :b  =  e  :  d,  show  that  d  :  //'  =  a'^  -{-  c^  :  b''  -^  d\ 

30.  A  bankrupt  owed  two  creditors  $1800.  The  sum  of 
their  credits  is  to  the  less  as  3  :  1.     What  did  he  owe  each? 

31.  Discuss  the  general  i)roblem:  To  divide  a  given  quantity 
iVinto  parts  proportional  to  the  given  numbers  ?/?,  ».  p,  etc. 

32.  Divide  the  number  N  into  three  parts,  .r,  1/  and  z,  such 
that  X  shall  be  to  1/  as  2  :  3,  and  z  to  the  difference  between  .r 
and  ?/  as  3  :  2. 

^;^.  The  speed  of  the  steamship  Scrvia  is  to  that  of  the  Both- 
nia as  13  to  10,  and  the  first  steams  5  miles  farther  in  8  hours 
than  the  second  does  in  10  hours.     AVhat  is  the  speed  of  each? 

34.  The  speed  of  two  pedestrians  was  as  4  :  3,  and  the 
slower  was  5  hours  longer  in  going  36  miles  than  the  '-ister 
was  in  going  24.     What  was  the  rate  of  each? 

35.  A  chemist  had  two  vessels,  A,  containing  acid,  and  B,  an 
equal  quantity  of  water.  He  poured  one  third  the  acid  into 
the  water,  and  then  poured  one  third  of  this  mixture  back  into 
the  acid.     What  was  then  the  ratio  of  acid  to  water  in  A? 

36.  If  24  grains  of  gold  and  400  grains  of  silver  are  each 
worth  one  dollar,  what  will  be  the  weight  of  a  coin  containing 
equal  parts  of  gold  and  silver  and  worth  a  dollar? 


noo 


RATIO  AND  PRO  PORT  I  ON 


w 


37.  Wlitit  common  quantily  juust  l)c  subtraclod  from  tlio 
four  (jiiiuititics  m,  «,  x  and  //  tluit  tlic  rt'iiiiiiiidors  nuiy  form 
H  i)ro[)ortioii? 

38.  A  chemist  has  two  inixtiireH  of  alcoliol  and  water,  tlie 
one  containiiif,^  IH)  ))er  cent,  of  alcohol,  the  other  50  per  cent. 
How  much  of  the  lirst  must  he  add  to  1  litre  of  the  seconil  to 
make  a  mixture  contiiininf,'  80  per  cent,  of  alcohol? 

39.  It  is  a  law  of  mechanics  that  the  distaiuies  throui^h 
which  heavy  bodies  will  fall  in  a  vacuum  in  dilTerent  times  am 
pi-o[)ortiomd  to  the  sfjuares  of  the  times.  If  a  body  fall  -is 
feet  farther  in  2  seconds  than  in  1  second,  how  far  will  it  fall 
in  1  second?     How  far  in  t  seconds? 

40.  Find  jin  expression  such  that  if  you  subtract  m -f '^ 
and  m  —  /^  the  ratio  of  the  remainders  shall  be  n  :  m. 

41.  On  a  line  are  two  points  whose  distance  is  a.  The 
first  point  divides  the  line  into  parts  whose  riitio  is  2  :  3;  the 
second  into  })arts  whose  ratio  is  5  :  7.  What  h  the  length  of 
the  line? 

42.  If  a  lino  is  divided  into  two  parts  whose  ratio  is  m  :  n, 
what  is  the  ratio  of  the  length  of  the  whole  line  to  the  distance 
of  the  }>oint  of  division  from  the  middle  point? 

43.  A  lino  is  divided  into  three  segments  ])roi)o rational  to 
the  numbers  m,  p  and  q.  What  is  the  ratio  of  the  parts  into 
which  the  middle  point  of  the  line  divides  the  middle  segment? 

44.  Divide  $:3.sr)  among  three  persons,  A,  \\  ami  C,  so  that 
the  share  of  A  shall  be  to  that  of  B  as  ('»  .  11,  and  that  of  C 
shall  be  $oO  more  than  those  of  A  and  B  together. 

45.  A  sailing-ship  loaves  port,  and  12  houi's  later  is  fol- 
lowed by  a  steamshi}>.  Jf  the  ratio  of  the  si)eeds  is  W  :  S 
how  long  will  it  take  the  steamer  to  overtake  the  ship? 

46.  A  courier  started  from  his  post,  going  7  miles  in  o 
hours.  Two  hours  later  ajiother  followed,  going  7  miles  in  2 
hours.     IIow  long  will  the  second  be  overtaking  the  lii'st? 

47.  IMie  aroiis  of  the  openings  of  two  wator-faucets  jiro  in 
the  ratio  o  :  o;  the  sj)eefls  of  flow  of  the  water  thi-ougli  the 
openings  are  in  the  ratio  3:4.  At  the  end  of  an  hour  I'^^I 
gallons  more  have  flowed  through  the  second  than  through  the 
first.      Wliat  was  the  How  from  each? 


PlionLKMS. 


r)()7 


(0(1  from  the 
31*8  may  form 

1(1  water,  tlio 
•  50  per  cent, 
the  second  to 
(.1? 

n('(>s  t]irou,ij;li 
•cut  times  an! 
a  body  fall  4S 
far  will  it  fall 

il) tract  m  +  n 
n  :  m. 

CO  is  a.  Tlic 
0  is  2:3;  tht^ 
,  the  length  of 

ratio  is  m  :  n, 
to  the  distance 

•()])()riional  to 
tlie  parts  into 
ddle  segment? 
!ind  C,  so  that 
and  tliat  of  C 
ler. 

rs  later  is  fol- 
:))eeds    is  \\  \  S 


w 


ship 


cr  7  miles  in  o 
g  7  miles  in  '^ 
o  the  lirst? 
faucets  are  in 
r  through  the 
f  an  hour  I'^'-^l 
an  through  the 


,i 


48.  'rhe  (lows  from  two  fauccls  into  two  0(iual  vessels  is  in 
the  ratio  -l  :  7,  and  both  vessels  were  placed  under  them  at 
tlie  same  moment.  When  the  vessel  under  the  hir^er  faucet 
was  full,  it  waH  renu)ved  and  the  other  put  into  its  place.  In 
80  seconds  from  the  time  of  beginning  both  vessels  were  lillcd. 
How  long  would  it  take  each  faucet,  to  iill  one  of  the  vessels? 

4(;.   'riiree  numbers,  a,  b  and  c,  arci  so  related  that 


in  :  n, 
P   '  (/• 


Find  the  ratio  c  :  a  -{-  0.     Kind  a,  b,  and  then  a  -|-  b,  in  terms 

of  ('. 

50.  If,  in  the  preceding  ])roblem,  the  sum  a  -\-  b  -j-  <'  =  -V> 
express  each  of  the  numbers  a^  b  and  c  in  terms  of  N. 

51.  The  speeds  of  two  trains,  A  and  li,  arc  as  m  :  n,  and 
the  journeys  they  have  to  nuike  as  p  :  q.  It  took  train  B  i 
hours  longer  to  nnikc  its  journey  thtin  it  did  train  A.  What 
was  the  time  required  by  each  train  for  each  journey? 

52.  A  street-railway  runs  along  a  regular  incline,  in  consc- 
fpience  of  which  the  specils  of  the  cars  going  in  the  two  direc- 
tions are  as  2  :  3.  The  cars  leave  each  terminus  at  n  gular 
intervals  of  5  minutes.  At  what  intervals  of  time  will  a  car 
going  up  hill  meet  the  successive  cars  coming  down,  and  vice 
versa ? 

53.  The  same  thing  being  supposed,  two  cars  starting  out 
simultaneously  from  the  termini  meet  at  the  end  of  30  minutes. 
How  long  in  time  is  the  journey  for  each  car? 

54.  The  same  thing  being  again  supposed,  a  rider  gallops 
up  hill  at  sucli  a  rate  that  he  passes  the  successive  cars  going 
up  hill  at  the  same  time  that  they  meet  the  successive  cars 
coming  down,  so  that  every  time  he  passes  a  car  going  up  Ik; 
meets  one  coming  do\tn.  What  is  the  ratio  of  his  speed  to 
tliat  of  each  of  the  cars? 

55.  Give  the  algebraic  answers  to  the  three  preceding 
questions  when  the  ratio  of  the  speeds  is  m  :  n. 

56.  Three  given  points.  A,  B  and  X,  lie  in  a  straight  lino. 
A  and  V>  are  ttiken  as  base-     .  » 


points  from  which  distances 


B 


608 


AM 770  AND  PllOrOR'i'ION. 


i  I 


r 


I        ( 


arc  nieiisiircd.   Having  given 

Distance  AB  =  h, 
Distance  AX  =  x, 

it  is  rofjuircd  to  find  the  i)08iti<)n  of  a  fourth  point,  Y,  between 
A  an(i  r»,  8uch  that  we  shall  have 

AY  :  YH  =-.  AX  :  liX  =  x  :  x  ~  b. 

Do  tliis  by  finding  the  distance  of  Y  from  A  in  terms  of  // 
and  X. 

57.  Show  that  in  the  preceding  construction  we  liavo 

AY  ^  AX        A13 

58.  Sliow  tluit,  in  tlic  ])receding  problem,  tlie  })roduct  of 
the  distances  of  X  and  Y  from  the  middle  point  of  the  line 
AB  is  -l/j\ 

59.  If,  instead  of  the  jmint  X,  the  point  Y  is  given,  find 
the  distances  AX  corres])onding  to  the  following  values  of 
AY,  in  order  that  the  same  proportion  may  hold  true,  and 
explain  the  results  when  negative: 


{a)  AY  =  lL 
{ft)  AY  =  %  b. 


Ans.  ^'  =  :r  0. 
o 


b. 


(r)  AY  = 

(r^)  AY  =  (i  +  .r)/;. 


(0  AY  =  \  A. 
(^)  AY  =  \  A. 
(//)  AY  -  %  A. 

m  AY  =  -A. 

'  n 


Hemark.  WJien  foin*  points  on  a  straight  line  fulfil  tlic  preccdini; 
proportion,  they  are  called  four  liarmonic  points,  and  the  line  AB  is 
said  to  be  divided  harmonically. 

60.  It  is  a  theorem  of  mechanics  that,  in  order  that  two 
masses,  V  and  W,  at  the  ends  of  a  lover,  AB,  may  bo  in  equi- 
librium, the  distances  of  their  points  of  suspension,  A  and  B, 
from  the  fulcrum,  F,  must  be  inversely  pro2)Oiiio}ial  to  their 
weights;  that  is,  we  must  have 

Weight  V  :  weight  W  =  FB  :  FA. 


PnollLI.MS. 


U)\) 


Now,  if  the  IcMuitli  y\I')  of  tlio  lover  \a  /,  jin«l  the  wei'Mils  of 


A 


V,  between 

tornis  of  /' 
c  have 


])ro(luct  of 
,  of  the  line 

5  given,  find 
ig  values  of 
»ld  true,  ;uid 

^  i-  -• 

i 

t  '^  A. 

llic  preceding 
the  line  AB  is 


iler  that  two 
ly  be  in  cqui- 
on,  A  and  B, 
ional  to  their 


V  and  W  arc  respectively  w  and  n,  express  the  leiigihsAF 
and  FH  of  the  arms  of  tlie  lever. 

6i.  The  weights  at  the  ends  of  a  lever  are  8  and  13  kilo- 
graiiunes,  and  the  fulcrum  is  3  inches  from  the  middle  of  the 
lever.     What  is  the  length  of  the  lever? 

62.  The  sum  of  the  two  weights  is  25  pounds,  and  the 
ratio  of  the  distatu^e  of  the  fulcrun»  from  the  middle  point  to 
the  length  of  the  lever  is  2  :  0.     What  are  the  weiuiits? 

63.  '^rhe  weights  are  m  and  n  {ni  >  ;/),  and  one  iirni  of  tlu' 
lever  is  h  long*  r  than  the  other.  Express  the  lengtii  (»f  the 
lever. 

64.  A  lever  was  balanced  with  weights  of  7  and  0  kilo- 
grammes at  its  ends.  One  kilogramme  being  taken  from  the 
lesser  and  added  to  the  great-er  (making  the  weights  G  and  lU 
kilogrammes),  the  fulcrum  had  to  be  moved  2  inches.  \\  hat 
was  the  length  of  the  lever? 

65.  A  line  is  divided  into  three  ])arts  ])roportional  to  the 
numbers  3,  4  and  5.  What  is  the  ratio  of  the  parts  in  which 
the  middle  point  of  the  line  divides  the  middle  segnu'nt? 

66.  To  300  pounds  of  a  mixture  containing  2  parts  of  zinc, 
3  of  copper  and  4  of  tin  was  added  200  ]  ounds  of  another 
mixture  of  the  same  metals,  when  it  was  f(;und  that  tlie  ])ro- 
])ortions  were  now  as  3,  4  and  5,  What  were  the  proi^ortions 
in  the  mixture  added? 

67.  Find  two  numbers  whose  sum,  difference  and  product 
are  to  each  other  as  the  numbers  5:1:18. 

68.  Find  two  numbers  in  the  ratio  7  :  3,  the  ratio  of 
whose  difference  to  their  product  is  1  :  21. 


510 


RATIO  AND  PllOrORTION. 


I   » 


i  .f 


i        » 


69.  Find  two  numbers  such  that  the  first  shall  be  to  the 
second  as  their  sum  is  to  ?)\,  and  as  their  difference  is  to  ^5. 

70.  Find  three  numbers  whose  sum  is  73,  and  such  tliaf,  if 
2  be  subtra(3ted  from  the  lirst  and  second  their  differences  will 
be  to  each  other  as  1  :  3,  and  if  2  be  added  to  the  second  and 
third  their  sums  will  be  to  each  otiier  as  4  :  5. 

71.  Two  boats  start  in  a  race.  Tlie  second  boat  rows  25 
stroices  to  the  first's  28,  but  10  strokes  of  tiie  second  are  equal 
to  12  of  the  first.  If  the  distance  between  the  jjoats  at  starting 
is  30  strokes  of  the  second  boat,  how  many  strokes  will  it 
make  before  reacliing  the  first? 

72.  One  cask  contains  18  gallons  of  wine  and  6  gallons  of 
water;  another  contains  12  gallons  of  wine  and  18  gallons  of 
water.  How  much  must  bo  taken  from  each  to  form  a  mix- 
ture coiiiaining  8  gjJlons  of  wine  and  8  gallons  of  water? 

73.  Two  mixtures  of  wine  and  water  contain  respectively 
\  and  I  wine.  How  much  of  each  must  l)e  taken  to  form  44 
gallons  of  a  mixture  of  which  the  wine  is  to  the  water  as  5:0? 

74.  A  and  B  ran  a  race  in  G  minutes.  B  had  a  start  of 
20  yards;  but  A  ran  5  yards  while  B  ran  4,  and  won  by 
10  yards.     AVhat  was  the  length  of  the  race,  and  the  rate  of 


running.'' 


75.  A  jeweller  has  three  ingots  of  metal.  A  pound  of  the 
first  contains  7  ounces  of  gold,  3  ounces  of  silver  and  G 
ounces  of  copper;  a  pound  of  the  second  contains  12  ounces 
of  gold,  3  ounces  of  silver  and  1  ounce  of  copper;  a  pound 
of  the  third  contains  4  ounces  of  gold,  7  ounces  of  silver  and 
5  ounces  of  copper.  He  wishes  to  form  an  alloy  weighing  1 
pound,  which  shall  have  &  ounces  of  gold,  3f  ounces  of  silver 
and  4;^  ounces  of  copper.  How  much  must  be  taken  from 
each  ingot? 

76.  The  king  of  Syracuse  gave  a  g(ddsmith  10  pounds  of 
gold  with  which  to  make  a  crown.  When  it  was  finished  tlie 
king  gave  tlie  crown  to  Archimedes  to  ascertidn  if  it  was  pure 
gold.  The  philosopher  knew  that  gold  weighs  .948  as  much 
in  water  as  in  air,  and  silver  .901.  When  the  crown  wns 
weighed  in  water  he  found  it  lost  10  ounces.  What  was  the 
quantity  of  gold  and  silver  in  the  crown? 


ihall  be  to  tlie 
dice  is  to  25. 
id  such  that  if 
liircrences  will 
lie  second  and 


l)oat  rows  25 
cond  are  equal 
)ats  at  starting 
strokes  will  it 

id  6  gallons  of 
;1  18  gallons  of 
to  form  a  mix- 

of  water? 

ill  respectively 

ken  to  form  44 

water  as  5  :  (5  ? 
had  a  start  of 
i,  and  won  by 
and  the  rate  of 

A.  pound  of  ihf^ 
I  silver  and  0 
bains  12  ounces 
pper;  a  pound 
3s  of  silver  and 
Hoy  weighing  1 
Hinces  of  silver 
be  taken  from 

h  10  pounds  of 
iras  finished  the 
n  if  it  was  pure 
IS  .948  as  much 
the  crown  wns 
What  was  the 


IliliA TIONAL  EXl'lU•:s.^lO^\<. 
IrratioiiJil   Expressions. 


nil 


Execute  the  following  divisions: 


I. 

2.  27.?:«Z/"r'3  -^  \)aU^c. 

3.  12aS/yfi  ^U^-bh. 


."'- , 


4.    X'^'t/Zi  H-  X'st/'Z"  i. 


5.  a 

6.  X 

7.  a 


—  3 
j 

-  7  A  —  1 . 


a 


_  1  ;.  _  b 


-'z-\ 


8.  da'b~'c-'  -Iba'c-'-^Sa-'bc 

9.  24x'i/'z -{-Ibx-y z'-')xi/ 

1 0.  ^bx'yz - '  -  2lx\ij  -  h'  +  7// -'z' ~  ■ 

1 1 .  20x''y  -  'z'  -  4:y''z'  -  12a;  -'y-' ~  4.^ 

12.  2Sx'^yz-'  -f  Wxy-'zi  —  Ux-'iy-'z-'  -=-  Lv'iy 

13.  ai  —  aib^  -=r  a^. 

14.  x^  —  xUi^  +^''  -•-  ^'»- 

15.  12^^^  — 30f4  ~  I2ai. 

m 

16.  2x'^  —  (jx-''^2x\ 

n 

17.  82-^— 4.r*  ~  2x-K 


§182. 

Express  the  following  products  of  irrational  quantities 
with  a  single  fractional  exponent  by  reducing  the  fractional 
exponents  to  a  common  denominator,  and  then  reversing  the 
process  of  §  182 : 


X    V    Z  1 

Prove  the  equation  rtn/>«c»  =  {(fb^c'^Y. 
aibki.     Aus.  amri^  =  {n'b'c)K 
7)ihnhpl.  4.  24;U.     Ans.  \2k 

2K3i.  6.  aibi. 


I. 
2. 

3- 

5- 

j_  1 

8.   .?;"'//". 
II.  a77iK 


7.  hh'isi. 


9.  32ir?i/^^ 
12.   c'ib's. 


V^-TV5"^^G. 


10. 

13.  7*.  5i. 


14.    A-V; 


V3 


1 


2-i;ji. 


16. 


(^S 


12- i. 


iWfP 


ril2 


mil  A  TIONA  L   EXPJiBSSl02\S. 


f 


1" 


§183. 

Reduce  tlie  following  expressions  to  monomials: 

I.    VSi-]-  V2I+  ■'/'*}.       Ans.    4/0  (  4/9+ 1^44-1)=  GI/g! 

4.  V2-  Vi^  +  1^3:3.  5.  V75  +  V-is  -  i^a; 

6.  '/ili  4-  ^  /^+  3  1^5  -  9  t^48. 

7.  I^ir^  -  V\ki  +  f/25«.  8.    l/^Zx'  +  i^//^-  V7^:. 
licduce  the  following  to  their  simplest  form  and  factor: 

9.    Vl^l? -^  Vb{)a^\  10.   {hrhy  -  {c^hY. 

11.  (2'V'^>V)1  -  (4.5V^/V)i  +  (4.GVZ;V)*- 

12.  (54rr +  ''/'=')i  -  (IGrt^'-^Z/^'^i  +  (:2«*'"  +  »)J  +  (:3ri'V)i. 

13.  (c/'"V' -f  a\/)i. 


15 


2> 


i^+f/ 


16. 


184. 


14.    [{a  +  h)\x  +  ij)-\K 
7)1  —  nf  vfp 


m  -\-  n  \7u'  —  ;>>  ni n  -f-  n ' 


Multiply: 
I.   {c  +  hVT)  {c  -  a  V7).       2.   {m  +  \^i)  {m  -  VTi). 

3.  {am  +  'i  ^(i'  —  z){}i  —  VI  Va  —  z). 

4.  (4  +  3  i^'^)  (4  -  3  V^).      5.   (5  -  G>^  V-.^)  (5  +  Gm  ^2). 

6.  j.^[l-(7>  -  IKHI  +  '^{p_-jn 

7.  {Vp-\-q-^Vp-q){Vp-^q-Vp-q). 

8.  (rt  +  x^  +  yi)  («  -  ^i  +  7/^).^ 

9.  ■ 


Vm.  Vn 

— —  "4~    — '^^^^ 

V)i         Viii 


V 


n 


V 


)ll 


Aggregate  the  following  fractional  expressions  and  sim- 
plify when  possible: 


74       ai 
II.  —  +  — . 
a        r 

7)ih        mi        m 


12. 


14.  r  -  /  + 


f  r  -  i 


q  UADRA  TIC  EQ  UA  TfOyS. 


r)i:J 


Equations  of  the  Second  Degree. 


1-1)=  {jVii. 
4/125. 

-  V7^.. 

,11(3  factor: 

2rt"'c')^. 

mp 

'Zmn-\-  n 


m  —  Vn). 
:)(5  +  G7i  V2). 

0. 


sions  and  sim- 

"  {c+xy 


§g  195-'40'^. 


9>'' 


+  1- 


110. 


<..! 


<>./• 


('4. 


3.   x^  -  \^x  +  G  = 
5.  x''  +  ^x  =  7. 

7.   a;^  —  mx  =  —  w. 


4.    X'  -  o7.': 


—  _  :io 


20. 


6.  x'^ 


8.  2;  + 


8.<;  =  -  12. 
1         1 


15       72  -  Ct 


II. 


x 

4/^ 


Oo-* 


=  2. 


l-x 


10.  2;  —  rt.v; 


12.    2'    —  OV.i' 


6.C 


ab. 


18. 


[3.   3:6-* +  .r  =  7. 


14.  4.x 


3G  - 


4(> 


15- 


17. 


40 


X  ~  o 
2:c  +  3 


-,  4-  ^i  =  13. 


48 


105 


'ix 


a;  +  3       x  +  10 
6^-.    18.   {x  -  3)  (.r  ~  8)  =  0. 


10  —  a:       25  —  32; 

19.  {x  —  a)  [x  —  b)  =  0. 

20.  (.T  +  4)  {CC  +  1)  =  6  {x'  +  1)  -  8.r 

21.  3  {x'  -  1)  -  24  =  4  (2;  +  5)  {x  -  3). 


O  ^,2 


22. 


{x  -  2)  (32;  +  1)  =  10  -  (22;  +  1)  {x  -  3). 


X 


X 


-  3 


X 


-  8 


23- 

2  (2;  -  3)       X  -V              "'*•  2;  +  2       22;  +  10 

5               29            3 

'-5. 

2  —  2;       4  —  ox       2x' 

26. 

2; +2       _3    ,       10.r 
32;  (22;  -  1)       X  "^  42;^  -  1' 

27. 

^_1         22;  -3         ^'-^28          «        _3:  +  ^' 
.^      -~   X  —  I        2;  —  9"         2;  —  6       22;  —  a 

3                 2                 1 

'6a 


2a 


a  —  3x 


30.   6  ^/.r  +  -^^  --=  37 


32. 


31.    V3.C-  54-   V^2;  +  0  =  0. 


"P 


l 

1 

1 

) 

514 


33-  1« 


Wn'Jl  ONE   UNKNOWN  QUANTITT. 


5  —  X       9  -  3a: 


2 


a; 


4"  3a:. 


.a-h3    ,    16 
34.     -.,      +-1; 


;.r 


;i.6- 


0 


26 
0 


35.   14  +  3.r  -  -^—  =  2.f  + 


36.  - 


X  +  19 


X 


^'-  :?+i  + 


u:  — 
_  4  _ 

X 
x+  1 


7 

9.r  -  8 
2      ■ 

_  1^ 


3 


38- 


X 


rc  +  G 


J.6'    —    0 


:,;  4-  4        4.r  +  7        7  -  a:        , 

a;  +  11       ^       9  +  4a- 
40. =  7  — 


X 


X 


41.   (3.r  +  1)  (4.T  -  2)  =  (13.1-  +  7)  (5a;  -  3). 


42. 


.7;^  +  2a:'  -f  -  8 
x'  +  :c  -  C~ 


=  a:"  +  a;  +  8. 


43-   (:^"  -  1)  (.?^  -  2)  +  (a:  -  2)  (a:  -  4)  =  V, 


44. 


46. 


10 


a: 


8  - 


J_0 
^^fl 


a; 


+  2 


45- 


^Sa- 
^^2 


0  + 


30. 

20 
3a: 


X 


2a:  -  11 


X 


X 


o 


3a;  -  3 


47-   7.^----— .5::  =  4a:  + 


6 
3:r  -  6 


a: 


48. 


;o. 


+ 


X 


13 


x-\-  1 

S.rJ-  3 
2a;^"l 


a: 


49. 


a: 


a: 


a:  —  4 
aT^^l 


20 


+  it:;:.  =  ,T  +  1-    5 


5a:  4-  3    ,    2a:  -  3 


/tei^ 


a: 


a:-  1 


+ 


22' 


52.  1/4  +  a;  +  4/a;  =  3. 

53.  2a:  -  a-'  +  4/Oa:*  -T2;?r+"7  =  0. 

54.  4/.c^^3  +  4/.r -f  8  =:  5  f'aJ. 


55.    4/2a^-f  1  -f  V7a:  -  27  =  l^3a:  +  4; 


ITY. 


_7 

c  -  5* 


;-30. 


+  : 


20 


dx 


—  4 


Ix  ~  :> 


2U" 


=  0. 


Cf/-^i)7e.l77(7  Agi^^77^,v^, 


56.   ??  +  _ 

3         4a;  —  3 


57.    x-\-  V;„^' 


?£_:il  5        5;c  __  3 


515 


^ _  2xj~  4 

4 


58.      i/M^+     i/^^=.l^^ 

59.     ^+l/a:^:=r4^       8^  5^  __ 

i/^^        -;•  00.    -- t  —  1     , 

61.  Va;  4-  3  4-  V'T;^ 17       -, .  "^ 

+  ''~'  =  ^«-  «-^  ^-  +  *'r+^=.A. 

63.  -. L  9  4/^;* 


Vi)x  —  1 


^  +  ^2'^x^ 


X  -  4/;;)  :|." 


a; 


-  =  m. 


64-    -T- 


a:   — 


7a:  +  3 


6.C.    - 


•'^  +  4^        ^.» 


X'  ~  X 


X  — 


4/, 


2a:' 


66.  :ri  __ 


5a: 


3 


^-\-':^  ^tt^ 


51 


a;' 


67.   ^ 


8a:»  +  10 


21 


12 

j£l+ 4   _  3a:' 
Sa:"  —  4        "8~ 


^^-   ^iMr^_|_|/-^ 


4^ 
'9-    -  -  _ 


+    3  |/; 


16^_ 

^^x'^ 


Vx  -^l  |/, 


-_    Va-  +  5 


70. 


|/. 


a-  + 


o 


X 


+  ^  -  -^^J^rx^- 


^.^^  =  «  f/- 


a: 


'ni  +  ^^a: 


+ 


71.   -Srj}^  ^  a -{.  ex 

Jl_^+_1  __  a-  4-  3 

+  5a:  -I-  G  ~  ^ipg* 


4^^ 


T2. 


^■\-^x 


73. 


--HJr  _    2'  +  2 


a;' 


74. 


12 


-  400^2:  =  f/. 


75.   —  - 


8 


.^4-3. 


C 


2-c  -  ^        2a:  -3'     76.   3.6-'  --  4a-  =: 


516 


Q  UA  1)11  A  TIC  h'Q  UA  TIOJ^S. 


"% 


77.    ox     —    iX 


o 
9 


^x. 


79- 
81. 
82. 
84. 
86. 
88. 

90. 

91. 
92. 

93- 
94. 


78.    (.T  _  3)  (a:  -  4)  =  2. 


80 


l^.r  +  I 


^x  -  1 


\x  -  3 


2:^-a  +  i  =  «- 


5 


3.C  +  2 
5 


1^ 

2 


3 


a;  +  l 
1  __        7 


-  1  =r 


X 


\ 


x-1       ^       x  —  'S 

5.r  +  3        2x-\-  o 


83. 

.85. 
87. 


_1 3+  =  -  -^ 


1 


3 


7  "^  3  (a;  -  U) 


=  1. 


Gx  -{-  b       2x  -\-  5  _  .J 
r  ;5 


2a;  +  5       3(.r  +  2) 

3.6-  -  3    I    13  _  1  -  bx 

4a;  -  1  "^  ¥  ~  ^^c  +  3 


i-  '^- 


'dx  +  4 

4:c        X  —  5 


IX 


r  3 

42;+_7 
"'  19     • 


7x'  -x-b  VJ  +  4x'  +  4  =  42;  +  7. 
x'  —  Qx  +  13  +  Vx'  +  ijx  +  9  =  5. 


O7.2 


-  3./;  +  0  4/.*;^  -  re  +  7  = 
'6x  -  3  4/./:'  +  3.f  +  9  =  30  - 


•)9 


a;' 


95.  3.1-'  4-  7:c  -  31  =  .?;  +  Vx:"  -^  3x  +  7. 

96.  V: 


97- 


3.r  +  3  - 

3.T'"'  -f-  L  - 
37+T- 

Vx'  +  5 

-  \' 
-V, 


98.   V\ 

99.   3  4 


TOO.    Vx'  +  3x 4-  V-f'  +  .c  4-  3  =  4. 


'.r  -  7  - 

3. 

:^;'^  -  3   _ 

_  1 

x'  -  3 

3" 

4a:  4-1  = 

:  i^i;.# 

/9.c^  4-  4.J 

;  4" 

4-38. 


5  = 


—  o 


Factor  the  foil 
tity  us  will  make  the  ti 
iiig  the  same  quantity. 


§198. 

owing  expressions 
rinomia] 


by  adding  such  a  quan- 
i^ei'fect  square,  and  subtract- 


a 


a' 


'ML 


3Z'^     Add  46',  and  subtract  it;  then 


'Zab-^-h'  -W  =  {a -by  -4b' 


=  [{a  -  b)  -  2b]  [{a  -b)-\-  2bi. 


-  4)  =  2. 

( v]i-2y 

^'Vj:  -  1 


5  (2;  -  {)) 

2x  4-  5 


2x 
5  _^ 
3" 

:f -3  -  - 
to  4-  7 

"""19     • 

ling  sucli  a  qiian- 
are,  and  sub  tract- 
it;  then 
a-b)^  2b\, 


WITH  nvo    Uiy KNOWN  QUA^-rrriEs. 


.       3.      2^    ~    4:X   +    3. 

5.  «'  -  'ZOahc  +  G4/yV 
7.   //'  -f  2b y  ~~  Hb\ 
9.   ;^•'  +  (J.?;  -j„  5, 
II.    -.^•'  +  .,^^10^ 
13.   ^«'  -  2((d  +  Z/\ 
15.   :r^  -  G..-'  -  IG. 

19.   4«*  -  3r«'^^  4-  9>. 


517 


2-  -i'  -  2ax  ~  3n\ 
4.  .'•'  4-  8./7/  -  0^'. 
6.   y^-'  -  7^  4-  3. 
8.   4^,'  -  4ab  -  ]r,/>'. 
10.   Ga'  4-  5,//,  _  (;//^^ 

12.  ^/^4-9.r7/  +  8l//. 
14    rt'^  4- 4^//>  +  ;v/. 
16.  x'^Sx'f, 


2.   :r  — 


//    =    5. 


(§  ^07.)        Simultaneous  Quadratics. 

Two  Unka^own  Quantities. 

I.  X  4-y  =    7. 

x'  4-  //'  =  25. 
3.   2a:  -   3?/  =    1. 
3a;"--4^z/r=  15. 

5.  x'  +f  =  a\ 

x'  ~  f  =    h\ 
7.   x'  4-  y'  =  1G9. 

xy  =  60. 


II. 


•-^  +  «/  =      8. 

x'Jr  y'=  224. 

^  +  3/  +  i/^y  =  19. 
x'  4-  7/»  =  97. 


13.  ?/=•  =  4a-z/. 

a;     -     ?/     =r    15. 

15-  2x  4-  3?/  =  17. 
xy  =  13. 

17.  5a:  —  3j/  =  1. 


x'  -  2a:^  =  01^ 
4.  2x  ~   2y  =  5. 

5x'  ~  3xy  ~y'  =,  iGi. 
6.  X    -i-y  =    2S. 

'^-y  =  147. 
8.  a,-'  4-  ^'  =  224. 

xy  =  \  2. 

10.  i?:^  =  a^ 

3xy  -^-^x  +  y  =  485. 
12.   10a:  4-  ?/  r=  3^^,/, 

;/  =  2  +  :?:. 
14.  ^^•  4-    2//  r=  30. 

.y'  -  10.r  =  lOy  4-  36. 
16.   3a:  4-  5y=  31. 

xy  4-  2/"'  =  18. 


2;/'^  _   /-^ 


x'^  ~  3xy  4-  10a:  -  5^  =  1. 


1 8.  4a:      —  5//   =  1. 

lly'  ~  ox'  -  9xy  4.  22.r 


19. 


7a:'^  -  13xy  -{-  5f  =  _  5. 
6a;'  -  Oa-?/    4-  4.y'  =  6. 


-  7^'  =  20. 
20.    3a:'  —  llxy  -\-  ^y' 
^rx'  -  17xy  -j-lhf 


7. 
17. 


618 


Q  UADliA  Tli '  KQ  UA  TIONS. 


A 


21. 

5a;'       nxy  +7?/'=  5. 

Cvx'  -  Voxy  +  \)y'  =  15. 

23- 

X     -\-  y    =     ('}. 

X    +  \i   -^  7;>. 

25- 

a-    -f  //    =  5. 

J          1       35.r?/ 

27. 

a;    -|-  7/    =  xy. 

xy  =  .7;'  —   //*. 

29. 

^'''  +  y'  +  .'i  +  2/  =  18. 

a-^  =  G. 

31- 

1        1   _  5 

a;        y  ~  6* 

a:  +  7/   =  5. 

33- 

2:i  +  ?/i  =      6. 

xl-^yl:=  126. 

35- 

9 

^        x-y 

,      820 
^        xy 

37. 

X        5 

1        y  _20 
xy       X        3  * 

39- 

ex'  +  3?/  =  27. 

4a;''  -  .^'    =-}  J. 

41. 

x'    -i-y'   =  45. 

a;     =  2y. 

43- 

xy  =  12. 

3a;  -  2.?/  =  1. 

45. 

X    -\-  xy  =  ?4. 

y     4-  .r?/   =21. 

47- 

^'  4-  xy  =  35. 

i?/»  +  :77/  =  14. 

49. 

a;'  -  2a.7/  +  ?/'    =  7. 

22.   'S/'  -  20a7/  4-  7y  =  15. 

7a;''  -  iSu'y  +19^' =r  II. 
24.  a;     -f-  .V   =        8. 

x'   +  .?/'  =  33G8. 
26.  a;^         —  y^  =z  50. 

a;^  (.-c  —  y)  =  10. 

28.  xy  '\-  xy"*  =  18. 

a;    -|-  a-y'  =  '>7, 
30.  a;'   +  ^'    -  a-  -  y  =  78. 

2-?/  +  a;     -f-  y  =  39. 

32.  3/  -  2a;'  =  19. 

y*  {-  xy   =  15. 

34.  2a;'  +  3a7/  =  20. 
3y'  4-  ^xy  =  39. 


36.  .^'    -  2a7/  =  24. 


xy  —  2?/    =    4. 

38.  x-^y=  Vb-^2. 


X 


y 


a;'  -  3.ry  +  2?/'  =  -  2. 


40.  4a;'  -  5?/'  =  16. 

3a;'  +  2^'  =  35. 
42.  3a;  —  2y  =0. 

xy  —  X     =  8. 
44.  4a;'  —  by  =  4:xy. 

6x  +  3y  =  37. 
46.  a;'  +  ?y'  +  a;4-?/ 

a;'  —  ?/'  -j-  a;  —  ?/  =  24. 
48.  a-'  +  .r//  +  y'  =  7. 

a;'  —  xy  -{-  y''  =  3. 
50.  a;  -\-xy  +  ?/  =  11. 

^•"  +  ^^  +  2/^=19. 


36. 


f  ^'  = 

8. 

=  15. 

rz   11. 

;3G8. 

=  50. 

=  IG. 

18. 

X  -  y  =  '/S. 
y  =  39. 

=  19. 

=  15. 

=  2G. 
=  39. 

=  24. 

=    4. 
5 -f  2. 

/5: 

=  16. 
=  35. 
=  6. 
=  8. 
=  ^xy. 
=  37. 

X  -\-  y  =  ^^' 

X  —  y  =  24. 

y"  =  ^• 
?/'  =  3. 

f  =  19. 


WITH  TUIIEE  UNKNOWN  QUANTIllES.  C)\d 


r'y\ 


x'-{-2  _  19 

.y'+  2  ~   3'* 
53.   2-\-y    =40 

X  +  2?/  =  7. 
55.   x'  -  xy  =  35. 

xy-  y'  =  10. 
57.   .r//'  4-  :ry  =z  18. 

a^^^  +  .r    =  27. 
59.   .?•  -y    =    8. 

x'  -  ;/''  =  80. 
6r.   4(.i--f-2//)  =  12. 

x'      -  iy'  -  33. 
63.   a:    —  ?/    =    2. 

2;'  +  7/'  r=  34. 


Three  or  more 
I.  XII  ~  24. 
(7  -  .V)^  =  8. 
•        (3-.r)(.-li)  =3. 

3.  .r^,^  =  3  (.r' +  4)  r=  12 

4.  a:^  +  y'  +  z'  =  84. 

•'^  +  .y  +  ^  =  14. 
xy  =  8. 

6.  X  -\~  y  -}.  z  =  12. 
^y  +  y^  +  ^a;  =  47. 
x^  +  y'  -  2^  =.  0. 

S.   2.r'  +  2xy  +  ?/»  =  49. 
'-'•'  —  .T2;  +  ^'  =  28. 
y'  +  ^'/^  +  ^'  =  25. 

I  O.       X    4-    _y    -L    ;^     -    9. 

x'  J^fJ^z^^  29. 
!/'  =  4^  +  1. 


52.  a;'  +  //  =  9  (7-  -  7/). 

a;  +  y=  4  (.r  -  y). 

54.  .r'  -  .r^=:  14. 

x'  H-  //'  =  74. 
56.  x'  -  3.ry  -f  2?/»  =  1. 

.^•'-f  2.f^-  4//'  =  5. 
58.       i/.7+     <^=    5. 

X  Vx  +  ;/  /y  =  35. 
60.  4.r'  -  9//'  =  7. 

2.r  +  :Xy  =  7. 
62.   :r'  +  ;/"  =  25. 

'-^  +  //   =    7. 
64.   X  —  2//  =      2. 
x'  4-  4?/*  =  100. 

UxK>:o\vjf  Quantities. 

2.   .t'  +  .r^;  rr  24. 
z'  -j-xy  =  12. 


y'-j-yz-i-z 
(x  -{-z)=4.{x'  +  z 
_  5.  :c  +  y  -I-  2;  = 

xz  ~  y\ 

'±±Jl  _  5 
xy  0* 

x+  z  __  3 
rt'2;      ""  4 ' 

.y  +  ^  __7 

~1'2" 


=  ^8  -  3y. 
-  10). 

14. 
=  84. 


yz 
9.   .7;  —  ?/  -f  22; 
a;'  +  y'^  +  2'^ 

^//  =  ^  +  .?/  - 
II.    x-\-  y  -\-  z  = 

^'  +  ;y'  +  z' 

xy  +  2-2  =  X- 


—  0 

—    -v. 

=  49. 

-  3. 
10. 

=  38. 


r)2o 


g  UAUIiA TIC  J£(^  UA  TIONS. 


\       ♦ 


12.    X  -f-  //    =  'J'. 
U  -\-  V    =1. 

X  +  ?<'  =  8. 

y  -f  f '  =  4. 
14.   a:K  =  yv. 
x  +  7/    =  U. 

?+-!^=  4. 


13- 


15- 


a:  +  //  =  9. 
w  -\-  V  —  9. 
a;'  +  u'  =  52. 
y"  +  r'  =  41. 
a,w/  =  35. 
?^r  =  18. 
a--f-  ?<  =  13. 


y  +  '^ 


0. 


Problems  Loiidin^  to  Quadratic  Equatious. 

1.  A  i)rincip{il  of  $0000  umoiints  with  simple  interest  to 
$7800  after  a  certain  number  of  years.  Had  the  rate  been  1 
per  cent,  higher  and  the  time  1  year  longer,  it  would  have 
amounted  to  $720  more.     What  was  the  lime  and  rate? 

2.  A  courier  left  a  tov/n  riding  at  a  uniform  rate.  Three 
hours  afterwards  another  followed,  going  1  mile  an  hour 
faster.  Two  liours  after  tiie  second  another  started,  going  6 
miles  an  hour.  Th.;y  arrive  at  their  destination  at  the  same 
time.    -What  was  the  distance  and  rate  of  riding? 

Alls.  Dist.  =  00  or  G.     Speeds,  4,  5  and  6  or  1,  2  and  C>. 

3.  In  aright-angled  triangle  the  hypothenuse  is  5  and  the 
area  G.     What  are  the  sides? 

4.  Find  two  numbers  whose  product  is  180,  and  if  the 
greater  be  diminished  by  5  and  the  less  increased  by  3,  the 
product  of  the  sum  and  difference  will  be  150. 

5.  Find  two  numbers  whose  sum  is  100  and  the  sum  of 
their  square  roots  14. 

6.  Find  two  numbers  whose  sum  is  35  and  the  sum  of 
their  cube  roots  5. 

7.  By  selling  a  horse  for  $130  I  gain  as  much  per  cent,  as 
the  horse  cost  me.     What  did  I  pay  for  him? 

8.  What  is  the  price  of  apples  a  dozen  when  four  less  in 
20  cents'  worth  raises  the  price  5  cents  per  dozen? 

9.  The  sum  of  tlie  squares  of  three  consecutive  numbers  is 
149.     What  are  the  numbers? 

10.  If  twice  the  product  of  two  consecutive  numbers  be 
divided  by  three  times  their  sum  the  quotient  will  be  4*  What 
aj*e  the  numbers? 


i 


rnon/jj.ws. 


.021 


nuatious. 

le  interest  to 
c  rate  been  1 
it  would  luive 
tid  rate? 
rate.  Three 
nilo  an  hour 
arted,  going  C 
m  at  the  same 

V 

0  or  1,  2  and  0. 
is  5  and  the 


80,  and  if  the 
ased  by  3,  the 

id  the  sum  of 

id  the  sum  of 

ch  per  cent.  a> 


en  four  less  in 
ive  numbers  is 


,1 


Hve  numbers  he 
11  be  f    What 


11.  A  woman  bought  a  numiier  ( f  oran^'cs  for  DO  cents. 
If  slio  had  bought  4  more  for  tiie  same  money  she  would  have 
])aid  t  <->f  ti  cent  less  for  each  orange.     How  luauy  did  nhe  buy.' 

12.  In  mowing  GO  acn^s  of  grass,  5  days  less  wouM  iiave 
been  suflicient  iT  2  acres  more  a  day  had  been  mown.  Uow 
many  acres  were  mown  per  day? 

13.  A  broker  bought  a  certain  number  of  shares  (par  SlOO 
each)  at  a  discount  for  $0400.  When  they  were  at  the  same 
j)er  cent.  j)remium,  he  sold  all  but  ^0  for  ii<T::i00.  IIow  many 
shares  did  he  buy,  and  at  what  })ricc? 

14.  If  the  length  and  breadth  of  a  rectangle  were  each  in- 
creased by  2,  the  area  would  be  JilJH;  if  both  were  each  dimin- 
ished by  2,  the  area  would  be  130.  Find  the  length  and 
breadth. 

15.  Twice  the  product  of  two  digits  is  equal  to  the  number 
itself;  and  7  times  the  sum  of  the  digits  is  equal  to  the  number 
formed  by  the  same  digits  reversed.     What  is  the  number? 

iC\  The  sum  of  two  numbers  is  ^  of  the  greater,  and  the 
dilTerence  of  their  S([uarcs  is  45.     What  are  tiic  numbers? 

17.  The  numerator  and  denominator  of  two  fractions  are 
each  greater  by  2  than  those  of  another,  and  the  sum  of  the 
two  fractions  is  2|;  if  the  denominators  were  intercliangcd, 
the  sum  of  the  two  fractions  would  be  3.  What  are  the  frac- 
tions? 

18.  A  man  starts  from  A  to  go  to  B.  During  the  first  half 
of  the  journey  he  drive?  \  mile  an  hour  faster  than  the  other 
half,  J;  d  arrives  in  5f  hours.  On  his  return  he  travels  a  mile 
slower  uuring  the  first  half  than  wiien  he  went  in  going  over 
the  same  portion,  and  returned  in  6^  hours.  What  was  the 
distance  and  rate  of  driving? 

19.  A  person  who  has  $8800  invests  a  part  of  it  in  one 
enterprise  and  the  rest  in  another;  the  dividends  differ  in  rnfo, 
but  are  equal  in  amount.  If  the  sums  invested  had  exchang(Ml 
rates  of  dividends,  the  first  would  have  yielded  $200  and  the 
other  $288.     What  were  the  rates? 

20.  Divide  50  into  two  such  parts  that  their  product  may 
be  to  the  sum  of  their  squares  as  G  to  13. 

21.  A  company  at  a  hotel  had  $12  to  pay,  but  before  set- 


C>'J2 


(^ U AD II A  TW  F.q U.  1  TIO^S. 


\ 


tling  «J  left,  wlien  tlioso  romiiinin^  liiul  30  cents  apieco  more 
to  puj  tliun  before.     How  many  were  there? 

22.  A  (h'over  bought  a  number  of  sheep  for  $180;  after 
kee[)ing  10  i»o  sold  the  rest  for  $1*00,  and  gained  Mo^  cents 
a})ieco.     How  nuinydid  ho  buy? 

23.  Two  partners,  A  and  B,  gained  $140  in  speculation; 
A's  money  was  .'J  months  in  trade,  and  his  gain  was  $G0  less 
tbiin  liis  capital;  \Va  money,  which  was  $50  more  than  A's, 
was  in  5  months.     What  was  each  man's  caj)ital? 

24.  Divide  .'JO  into  two  such  parts  that  their  product  may 
bo  30  times  their  diiTercnco. 

25.  A  and  B  set  out  from  two  towns  which  are  VH)  miles 
apart,  and  travelled  until  they  met.  A  went  8  miles  an  hour, 
and  the  number  of  hours  they  travelled  was  3  times  greater 
than  the  number  of  miles  B  travelled  an  hour.  What  wero 
their  hourly  rates?  Ans.,  in  part,  B's  rate,  /aS  —  4. 

26.  In  a  purse  containing  538  ])icces  of  silver  and  nickel, 
each  silver  coin  is  worth  as  many  cents  as  there  are  nickel 
coins,  each  nickel  is  worth  as  many  cents  as  there  are  silver 
coins,  and  the  whole  are  worth  $1.50.  How  many  are  there 
of  each  ? 

27.  Find  two  such  numbers  that  the  product  of  tlieir  sum 
and  difference  may  be  7,  and  the  product  of  the  sum  and  dif- 
ference of  their  squares  may  be  144. 

28.  A  grocer  received  an  order  for  12  pounds  of  sugar  at 
12  cents  a  pound.  If  he  should  have  none  for  that  price,  ho 
was  to  send  as  many  pounds  more  or  less  tlian  12  as  the  sugar 
cost  less  or  more  than  12  cents  a  pound.  The  bill  amounted 
to  $1.35.  IIow  many  pounds  had  he  sent,  and  what  was  the 
price  per  pound? 

29.  A  grocer  sold  50  pounds  of  pepper  and  80  pounds  of 
ginger  for  $26;  but  he  S')ld  25  ]>()unds  more  of  pepper  for  $10 
tiiau  lie  did  of  ginger  for  $4.  What  was  the  price  per  pound 
of  e;ich  ? 

30.  A  and  B's  shares  in  speculations  together  amounted  to 
$075.  A  had  his  money  invested  5  months  and  B  \\  months, 
and  each  receives  in  cajutal  and  profits  $455.  What  did  each 
begin  with? 


I'liOIiLh'MS. 


rm 


apiece  more 


31.  A  person  rents  Ji  rertuiti  nmnbor  of  nores  of  land  fui* 
ItTiU;  lie  retains  10  iiercs,  and  suldcts  the  rest  at  W  vvnta  an 
ucro  more  tiian  he  }jjave,  and  receives  ^\'-l  more  than  lie  \m\n 
foi-  thu  wliole.  How  many  acroa  were  there,  and  how  niueU 
per  aerc? 

32.  A  person  bonglit  a  certain  nr.niber  of  shares  for  as 
many  dollars  per  share  as  the  nninher  he  buys;  after  they  rose 
as  many  cents  per  share  as  he  iiad  shares,  he  sold  them  and 
gaine(l  $4.      How  nniny  shares  did  lie  buy? 

;^;i.  The  income  of  a  certain  railway  company  wouhl  justify 
a  dividsnd  of  5  per  cent,  of  the  whole  stock;  but  as  ^150,000 
of  the  stock  is  prefe^'red^  guaranteeing;  (I  [)er  cent.,  the  divi- 
dend for  the  remaining'  stock  is  reduced  to  4|  per  cent. 
What  is  the  wliole  amount  of  stocJv? 

34.  The  length  of  a  rectani^uhir  farm  is  to  its  width  as  t 
to  .3;  I  is  in  grass,  and  the  remaining  45  acres  is  cultivated. 
What  are  the  dimensions  of  the  field? 

35.  If  a  straight  lino  be  divided  into  two  sucli  parts  that, 
the  rectangle  contained  by  the  whole  line  and  one  ])art  ise(|ual 
to  6  times  the  square  of  the  other  part,  what  will  be  tiie  ratio 
of  these  two  parts? 

36.  Out  of  a  sj)here  of  clay  whose  diameter  is  10  inches, 
two  spheres  arc  formed  with  radii  of  3  and  5  inches  respec- 
tively. If  the  volumes  of  spheres  vary  as  the  cubes  of  their 
radii,  what  will  be  the  radius  of  the  sphere  that  can  be  made 
of  the  clay  that  remains? 

37.  The  two  digits  of  a  certain  number  differ  by  1,  and 
their  product  is  \  of  the  next  higher  numl)er,  what  is  the 
number? 

38.  Find  five  numbers  having  equal  differences,  and  such 
that  their  sum  shall  be  40,  and  the  sum  of  their  cubes  3520. 

39.  A  merchant  bought  a  barrel  of  wine  for  $(50;  he  re- 
tained 13  gallons  for  his  own  use  and  sold  the  rcnuiinder  at 
an  advance  of  80  per  cent,  on  each  gallon  and  giiined  2<)  per 
cent,  on  the  whole.     At  what  price  per  gallon  did  he  sell  it? 

40.  Find  two  numbers  that  are  to  each  other  jis  9  to  7; 
and  the  square  of  tl  eir  sum  is  equal  to  the  cube  of  their  dif- 
ference. 


524 


q  UA  DRA  TW  KQ  UA  TIOAS. 


■  f 


41.  The  panel  in  a  door  is  12  by  18  inclies,  and  it  is  to  he 
SHiTounded  by  a  margin  of  uniform  width  and  equal  surface 
to  the  panel.     How  wide  must  tiie  margin  be? 

42.  The  fore  wlieel  of  a  coach  makes  G  more  revolutions 
than  the  hind  wlieel  in  going  ICiO  yards;  but  if  the  circunifei-- 
cnce  of  each  wheel  be  increased  bv  4  feet,  the  fore  wheel  will 
make  only  4  more  revolutions  in  IGO  yards.  What  is  the  cir- 
cumference of  each  wheel? 

43.  Tiie  sum  of  three  numbers  is  15;  the  difference  between 
the  lirst  and  third  is  3  more  than  the  difference  between  the 
second  and  third,  and  the  sum  of  their  squares  is  93.  What 
arc  the  numbers? 

44.  The  product  of  two  numbers  is  15,  and  if  their  differ- 
ence be  added  to  the  difference  of  their  squares  tlie  sum  will 
be  18.     What  are  the  numbers? 

45.  A  certain  number  consists  of  two  digits;  the  number 
is  4  times  the  sum  of  its  digits;  and  3  times  the  number  is 
equal  to  twice  the  square  of  the  sum  of  its  digits.  What  is 
the  number? 

46.  Find  two  numbers  whose  sum  is  14,  and  if  their  prod- 
uct be  added  to  the  sum  of  tlieir  squares  the  result  will  be 
1-18. 

47.  Two  brokers  begin  business  with  a  joint  capital  of 
810,000.  A  withdraws  at  the  end  of  12  months  and  receives 
1^4900  in  capital  and  profits.  B  remains  3  months  longer  and 
receives  $7800  stock  and  gain.  What  was  the  oi'iginal  capital 
of  each  ? 

48.  Find  five  equal  numbers  whose  sum  is  equal  to  their 
continued  product. 

49.  A  jockey  bought  a  horse  and  sold  it  at  a  certa'u  per 
cent,  profit;  with  the  money  he  bought  another  horse  and 
sold  it  at  the  same  per  cent,  profit,  and  with  lli(>  proceeds  he 
was  able  to  buy  2  horses  each  costing  2  ])er  cent,  less  than  the 
first.     What  i)er  cent,  did  he  make  on  each  transaction? 

50.  Two  travellers  start  from  the  same  place  at  the  same 
time,  one  goes  due  north  IG  miles  a  day,  and  the  other  due 
cast  21i^  miles  a  day.  How  long  must  they  travel  in  order  to 
be  IGO  miles  apart? 


%3 


PROBLEMS. 


x\\(\  it  is  to  he 
equal  surface 

re  revolutions 
tlie  circuiufer- 
'ore  wheel  will 
liat  is  the  cir- 

irence  between 
3e  between  the 
i  is  93.     What 

if  their  diffor- 
iS  the  sum  will 

s;  the  number 

the  number  is 

limits.     What  is 

i  if  their  prod- 
result  will  be 

)int  capital  of 
IS  and  receives 
Ulis  longer  and 
original  capital 

3  equal  to  their 

it  a  certa'n  per 
ther  horse  and 
li(>  proceeds  he 
it.  less  than  the 
msaction? 
iice  at  the  same 
the  other  due 
ivel  in  order  to 


51.  What  is  the  length  of  a  side  of  a  square  wliose  area  is 
increased  by  J  of  its  amount  when  4  feet  is  added  to  each  side? 

52.  Find  the  length  of  the  side  of  a  square  such  that  (lie 
number  of  square  feet  in  its  area  exceeds  the  number  of  linear 
feet  in  its  perimeter  by  12. 

53.  The  perimeter  of  a  rectangle  is  34  feet;  if  its  length 
were  increased  by  4  feet,  while  its  perimeter  remained  the 
same,  the  former  area  would  exceed  the  doui)Ie  of  the  second 
by  G  feet.     What  were  the  original  dimensions? 

54.  If  3  feet  be  taken  from  one  side  of  a  rectangle  whose 
perimeter  is  14  feet  and  added  to  the  other  side,  the  area 
would  be  doubled.     What  were  the  first  dimensions? 

55.  A  man  invests  his  money  at  a  certain  rate  of  interest 
for  two  years,  and  finds  that  ho  will  get  1  per  cent,  more  for 
it  if  he  reckon  by  compound  interest  compounded  annually 
than  by  simple  interest.     What  is  the  rate  of  interest? 

56.  A  person  bought  a  certain  number  of  shares  when  (liey 
were  at  a  discount  and  sold  them  when  they  rose  to  a  premium 
of  the  same  rate  ])er  cent.  His  profit  on  the  first  investment 
was  Ji  percent,  more  than  the  common  value  of  the  premium 
and  discount.     What  was  the  latter  and  the  rate  of  profit? 

57.  A  regiment  of  2196  soldiers  is  formed  into  two  scpiare.^, 
one  having  G  more  men  on  a  side  than  the  other.  How  many 
men  are  there  on  a  side  of  each  square? 

58.  Find  two  numbers  who.^e  })roduct  is  twice  their  sum, 
and  the  sum  of  their  squares  45. 

59.  Find  two  numbers  whoso  product  is  8  times  their  dif- 
ference, and  the  dilTerence  of  their  squares  48. 

60.  Find  two  numbers  whose  difference  is  G,  and  |  of  their 
product  is  equal  to  the  square  of  the  less. 

61.  Find  two  numbers  such  that  their  product  added  to 
twice  the  square  of  the  greater  is  G5,  and  the  product  aihied 
to  the  square  of  the  less  is  24. 

62.  Find  two  numbers  such  that  their  sum  multiplied  by 
the  sum  of  their  squares  is  715,  and  the  difference  multiplied 
by  the  difference  of  their  scpiares  is  99. 

63.  Two  trains  start  at  the  same  time  from  two  towns  and 
run  at  a  uniform  rate  towards  the  other  town.     When  they 


m^ 


rm 


PROGRESSIONS. 


f 


h 


meet  it  is  found  that  one  train  has  travelled  00  miles  more 
liian  the  other,  and  that  if  they  continue  Jit  the  same  rates 
tiiey  will  linish  the  journey  in  G  and  13^  hours.  What  arc 
the  distance  and  rates? 

64.  A  man  receives  $2200  a  year  interest.  If  he  liad  in- 
vested his  capital  at  I  per  cent  higher,  he  could  have  lessened 
nis  investment  by  $4000  and  received  the  same  income  as  be- 
fore.    How  much  had  he  invested? 

Progressions. 

Note.— The  abbreviations  A.  P.,  G.  P.,  C.  D.,  aiul  C.  R.  arc  but  for 
Aritliinelical  Progression,  Geometrictvl  Progression,  Conunon  Difference, 
and  Common  Ratio,  respectively. 

1.  If  the  first  and  last  terms  of  an  arithmetical  progression 
are  a  and  I  and  the  number  of  terms  n,  express  the  sum  of 
all  the  intermediate  terms. 

2.  If  the  first  and  last  terms  of  an  A.  P.  are  4  and  28 
respectively,  what  possible  values  may  the  sum  of  the  inter- 
mediate take? 

3.  Sum  to  n  terms  distinguishing  the  cases  when  n  is  even 
and  odd,  when  necessary: 

1-3+5-7+.... 

4.  2-4  +  6-8+.... 

5.  p^p  +  n,p  +  %n, 

6.  If  the  square  of  the  fourth  term  of  an  A.  P.  is  equal  to 
the  ))roduct  of  the  first  and  sixth,  show  that  the  tenth  term 
must  vanish. 

7.  If  the  square  of  the  second  term  of  an  A.  P.  is  equal  to 
the  product  of  the  first  and  fourth,  show  that  the  square  of 
the  sixth  is  equal  to  the  product  of  the  fourth  and  ninth. 

8.  Oeneralize  the  preceding  result  by  showing  that,  in 
order  that  the  square  of  the  nth.  term  may  be  equal  to  the 
])roduot  of  the  first  and  ;^'th,  and  the  square  of  the  mi\\  to 
the  product  of  the  ?/th  and  m'lh,  it  is  necessary  and  sufficient 
that  m,  m\  n  and  w'  fulfil  the  conditions 

m'  =  2  {)n  —  )i)    +  1;     :.'/.''  —  /ii  +  u 

9.  Find  three  quantities  in  A.  P.  whose  sum  shall  be  3a 
anil  the  sum  of  whose  squares  shall  be  \l(i\ 


PEOGIiESisIOyS. 


r)27 


hen  n  is  even 


lu  ^^luiU  be  3(1 


10.  Find  7  terms  of  an  A.  P.  such  that  their  sum  shall 
be  14  and  the  sum  of  their  squares  84. 

1 1.  In  an  A.  P.  the  product  of  the  first  and  eighth  terms  is 
less  by  k  than  the  product  of  the  second  and  seventh.  How 
much  less  is  the  product  of  the  third  and  sixth  than  that  of 
the  fourth  and  fifth? 


Expi 


th 


of 


12.  i^xpress  tne  sum  ol  n  terms  of  an  A.  P.  in  terms  of 
the  first  term  and  the  C.  D. 

13.  It  a  and  b  are  the  first  two  terms  of  an  A.  P.,  express 
the  last  term  and  the  sum  of  71  terms. 

14.  Prove  that  if  the  sum  of  m  terms  of  an  A.  P.  be  w, 
and  tiie  sum  of  71  terms  be  w,  we  siiall  have 


2  {m  -\-  7i)  -\-  tmid  =  0. 
15.  If  a",  F,  c*  be  in  A.  P.,  then, 


1 


a  -\-b'     c  -\-a'     b  -\-  c 

will  also  be  in  A.  P. 

16.  The  sum  of  the  first  three  terms  of  an  A.  P.  is  15  and 
the  sum  of  their  squares  is  83.     What  is  the  sum  of  n  terms? 

17.  In  a  progression  of  9  terms,  the  third  term  is  10  jjid 
the  sum  153.     Find  the  first  term  and  common  difference'. 

18.  In  an  A.  P.  a  certain  term  is  ^;  there  are  '2n  terms 
before  k  and  n  terms  after  it,  and  the  sum  of  all  the  terms  is 
'6n  +  1.     Find  the  C.  D. 

19.  Two  men  start  simultaneously  from  the  same  point  in 
the  same  dir'^ction.  The  one  walks  711  miles  the  first  dfiv, 
and  diminishes  his  walk  bv  h  miles  each  dav;  the  other  walks 
n  miles  the  first  day,  and  increases  his  walk  //  miles  each  day. 
How  far  will  the  latter  be  ahead  at  the  end  of  i  days? 


20.  Express  the  sum  of  the  G.  P.'s: 

a''  +  f/^"  +  rr^"*  +  .  .  .  . 


1   +  v/3  +  3    + 


+  3* 


lOw, 


21.  The  sum  of  the  first  and  seventh  terms  of  a  (1.  P.  is 
li,  and  the  sum  of  the  second  and  eighth  is  k.  Find  the  first 
term  and  the  C.  R. 

22.  The  sum  of  the  first  and  fifth  terms  of  a  G.  P.  being 


I' 


^l!li'  *' 


i  I 


'  I 


528 


PROGRICSSWNS. 


added  to  twice  tlie  third  term  gives  a  sum  which  is  0  times 
tlic  lii'st  term.     Find  the  C.  E. 

23.  The  fifth  term  of  a  G.  P.  exceeds  the  first  by  IG,  and 
the  fourth  exceeds  the  second  bv  4  V'i.  Find  the  first  term 
and  C.  R. 

24.  In  a  Ct.  p.  the  sum  of  71  terms  is  *S'and  the  sum  of  2?i 
terms  is  dS.     Express  tlie  C.  R.  and  first  term. 

25.  In  a  G.  P.  of  2n  +  1  terms,  whose  first  term  is  5,  tlie 
sum  of  the  first  and  hist  terms  is  125  greater  than  twice  ihe 
middle  term.     Find  the  C.  R. 

26.  The  first  term  of  a  G.  P.  is  2,  and  the  continued 
product  of  the  first  5  terms  is  128.     What  is  the  C.  R.  ? 

27.  Find  tliat  G.  P.  of  wliich  the  product  of  tlie  first  and 
second  terms  is  3,  and  that  of  the  third  and  fourth  terms  is  48. 

2<S.  A  })erson  who  each  year  gtiined  half  as  much  again  as 
l)e  did  the  year  before,  gained  $2050  in  7  years.  What  was 
his  gain  the  first  year? 

29.  A  man  who  had  a  principal  out  at  5  per  cent,  per 
annum  compound  interest  for  4  years  found  that  the  interest 
gained  during  the  second  and  fourth  years  was  greater  by 
$84.10  than  that  gained  during  the  first  and  third  years. 
Wiuit  was  the  principal? 

30.  Show  that  \ia,h,c,d.  .  .  .  k,  I  bo  in  G.  P.  we  shall  have 
{a  ^  b  -{-  c  -\-  .  .  .  .  -\-  k){b  -\-  c  +  d  +  .  .  .  .  -\-l) 

=  i(/;  +  f +  .Z  +  . ..  .  +  /r. 

31.  If  a,  b,  c,  d  be  in  (1.  P,  prove  that 

(«'  4-  //  4-  c')  {b'  -h  (■"  +  d')  =  {ah  4-  'r  +  cdy 

(/,  _  ry  -f  {c  -  ay  +  {d  -  by  =  {a  -  dy. 

32.  Gene»"aii/,e  the  Orst  of  the  preceding  results  by  allow- 
ing that  if  we  multiply  the  sum  of  the  squares  of  the  lirsi  // 
t<'i"ms  of  a  G.  P.  by  the  sum  of  the  squares  of  the  n  ternjs 
following  the  first  term,  the  ]U'odnct  will  be  equal  to  the 
s(iuare  of  the  sum  of  all  the  products  formed  by  multiplying 
each  term  from  the  first  to  the  nth  by  the  term  following  it. 

^^.  Sum  to  71  terms 


7N 


in 


)ll       ; 

m 


Hi 


ch  is  9  times 

st  by  IG,  and 
:he  first  term 

he  sum  of  2n 

:erm  is  5,  the 
luu  twice  the 

be  continued 
3  C.  R.  ? 

the  first  and 
h  tertns  is  48. 
nucli  again  as 
!.     What  was 

per  cent,  per 

it  tbe  interest 

as  greater  by 

third   vears. 

.  we  shall  have 

. . .  +  0 


-  </r. 

;ults  by  show- 
of  tbe  lirsi  u 

C  the  ti  tevms 
equal    to  th(^ 

)y  multiplying 
following  it. 


PJcOGiih:8sioys. 


529 


34.  In  a  G.  P.  of  0  terms  are  given: 

The  sum  of  all  the  terms  except  tiie  first  =  33- 
The  sum  of  all  the  terms  cxcei)t  tiie  hist  —  —'22. 
Find  the  series. 

35.  Find  two  quantities  of  wiiicb  the  Jirithmetioal  mean  is 
a  and  the  geometrical  mean  is  ^,  and  ])rove  the  result. 

36.  In  a  G.  P.  of  8  terms  the  product  of  tiie  four  alternafo 
terms  beginning  with  the  first  is  1,  and  the  product  of  the 
four  alternate  terms  from  the  second  to  the  eightli  is  10. 
Find  the  progression. 

37.  A  party  of  m  persons  have  .«?  dollars  unequallv  divided 
among  them.  Each  simultaneously  divides  his  moiu-y  equally 
among  his  m  -  1  fellows.  If  one  of  the  partv  had  a  dollars 
in  the  beginning,  how  much  will  he  have  after  1.2,  and  p 
such  divisions.^  ' 


s 


Ans.  — 

?n       111  —  1  \/// 


"V.    i--J.    -if 


(£-")^ 


Find  the  limits  of  the  sums  of  the  progies:iiuns: 
4  3 

38.  --  +  ,  +  _  +  .... 


■'°-  '  +  «+,?-+■■■• 


71    '   n 

42.    I  -f  (r  + 


43-    1  -    r-f 


11 


+  1''  +  ^)"%... 


1  \-' 


-f    r  + 


44.   r-^{\^ay^{\-^a^n'y^{\-^a\-a'-\-a')y'-{- 


r  and  ar  being  each  less  th; 


m  unitv 


45.  r-\-{\-a)r'\-{\-n^i-,r)r'^{\-a  \-(r -a')r' -[■ 


530 


FUNCTION  A  L   NO  TA  TION. 


\l  .1 


46.   r-{\-ny^{\-n-\-a'y-{\-a-^n'-a'y+ 

^7  1  :  '^  - 1 4.  ('^  -  ^y  , 


48. 
49. 


7i 


n 


71 


I  +  (»_:ii)!  +  ("  -  !)•  , 


n  -  1      (/i  -  1)'     {11  -  1) 


+ 


M  +  1    (^'-f-ir    O'  +  i) 


?i* 


50.  ?i + --— -  + 


/r 


r,  +  ■  .  •  . 


M+1  '  {n^\y 

51.  r+(a+i{»)rHK+^//>4-'^')'-'+('''+«'*+«^''+^')^-*+.. 


I       I 


Functional  Notfition. 

Prove : 

1.  {^n)\  =  3"  (1.3.5  ....  2m-  1)  .;/! 

2.  (2^)!  =  2"  (1.;J.5  ....  15)  (1.3.5.-;)  (1.3). 

Using  the  notation  [w]  =  1.3.5.7  ....  ;// 

k  =  2'* 
Sliow  that  we  have 


3.  ^•:==  2^-' [/;-!] 


1-' 


"I  -  1] [3J. 


^4u 

4-  lr> 


'2;A  _  [4;/  -  IJ 

'ji/     \  )i  I  ~~  [2h,  —ly 

5.  If  S{n)  represent  the  sum  of  the  first  u  terms   of   a 
geometrical  progression  wliose  C.  U.  is  7',  show  that 

,S'(2;/)  =  (/•"  +  1)<S'(>0- 

r 


.<^{A»)  =  {r"-\-l)i^^-^^j.S{n). 


6. 
denoc 

What  will  l>e  tlie  last 
ttinators  of  the  following 

V  f'^-^V  f'^  +  ^V 

i/'    \    .s    /'    \    s    /' 

(     »    \       ft  —  .«  /'A 

W+  1/  ~~  .s  +  1  vW' 

/;A        /     w     \  _  jn  — 
[jl  ^    [.^  +  1/        U  -1 

factors  in  tl 
cxi)ressions: 

le 

numerators 

and 
3\ 

7- 
8. 

+ 1/  \  5 

/ 

FUNCTIO  NA  L  NO  TA  TION. 


631 


y~v  — 


+/.');•* -f... 


;}). 


.  [3J. 


terms   of  a 

[Kit 


n orators  and 


If  >S'„  represent  the  sum  of  tlie  first  n  natural  ninnberjj, 
that  is, 

'^n  =  1  +  3  -f  3  + +  /i, 

show  tliat: 

9-  'S'„  +  ^S'„,,  =  («+  1)'. 

10.  Sn  :    «S'„  + 1  =  /^  :  ;?  -f-  2. 

11.  .%  X  *S;  X  /S'e  =  3!   [7J. 

12.  S.  X  8^  X  *S;  .  .  .  .   X  *%„  =  ;/!  [2;i  +  1]. 

13.  'S^  X   '\  X  >S\  .  .  .  .  X  iS^„  +  ,  =  {n  +  1)!  [;>;*  +  1]. 
14-  *^'«  X  *Ss  X  Si X  *S„  =  (27i  +  1)  (/i.7  l^n  -  \]\ 

15-  ^svf  >s;  +  6',  + -i-x,^ 

16.  S\  +  S,  +  S,-i- +  *%ufi 

=.  r  4-  3'  +  5^  +  .  .  .  .  +  {In  +  I)'-'. 

17.  If  d  =  k  +  56;_i  find  tlie  values  of  C^,  ('„  C\,  and  (\ 
in  terms  of  //,  .s-  and  Co,  and  find  tlie  value  toward  which  (\ 
approaches  as  i  increases  indefinitely,  assuminfr  s  <  1. 

18.  Apply  the  preceditii;  notation  to  the  following  problem: 
A  person  having  a  full  and  an  empty  cask  pours  half  the  con- 
tents of  the  full  one  into  the  otlior;  thou  half  of  this  last  one 
back  again.  lie  rci)eats  this  double  operation  an  indefinite 
number  of  times.  Find  what  fraction  of  the  liquid  Avill  re- 
main in  the  first  cask  after  1,  2,  3,  4,  and  i  such  double  opera- 
tions. 

To  do  this  assume  that  d  and  1  —  d  ropresont  tlic  frarlions  of  tho 
liquid  \\\  tlie  two  casks  after  the  ith  ope  ratio  u,  and  then  fuul  the  fractions 
alter  the  (t  +  l)st  operation. 

19.  A  vintner  has  one  cask  containing  a  gallons  of  wine 
and  another  containing  b  gallons  of  water.  He  pours  half  tho 
wine  into  the  water,  then  half  that  mixture  back  into  tho 
wine,  and  so  on  iiulefinitely.  Find  an  ox))rossion  for  tho 
quantities  and  proportions  of  wine  and  water  in  each  cask 
after  'In  and  also  after  2m  -j-  1  such  o])oiations. 


r)'A2 


p/niMiri'A Ti().\s  A .\n  coMnLwi  rio^\s. 


l*«riiiiiiatioiis  and  Combinations. 


1.  A  regular  cube  is  to  have  its  sides  numbered  1,  2  ....  T,. 
In  liow  many  ways  may  the  numbering  be  done? 

2.  In  how  many  ways  might  the  numbering  be  done  in 
the  hist  ])roblem  if  only  three  of  the  six  sides  were  to  be 
numbered? 

3.  A  party  of  3  boys  aiul  -l  girls  has  to  walk  in  single  file, 
the  boys  ahead.     In  how  many  ways  can  they  be  arranged? 

4.  Wiiat  would  be  the  numl)er  of  arrangements  in  the  last 
])roblen.  f  th  ;ily  condition  were  that  the  boys  must  be  to- 
gether i)     nir  p:">up  and  the  girls  in  another? 

5.  If  ti.i'3  €r;ju/;ination  of  any  three  dilferent  letters  in  any 
order  made  a  word,  ,.  ow  many  words  of  thr(.ie  letters  could  be 
formed  from  tlie  2G  letters  of  the  alphabet? 

6.  If  in  the  last  problem  the  words  thus  formed  were 
divi<led  into  sets  such  that  the  different  words  of  a  set  should 
be  formed  of  the  same  letters,  how  many  sets  would  there  be, 
and  how  many  letters  in  a  set? 

7.  Six  men  with  their  wives  arc  to  stand  in  a  row.  In 
how  many  ways  may  they  De  arranged  subject  to  the  condition 
that  each  man  must  remain  alongside  his  wife? 

8.  What  would  be  the  answer  to  the  last  problem  in  case 
each  man  had  to  keep  his  wife  on  his  right? 

9.  A  boy  has  the  letter  blocks  which  form  the  words  you 
are  mad.     In  how  many  of  the  arrangements  will  all  three 
words  be  recognized,  supposing  that  any  word  may  be  rocog. . 
nizedwhen  its  lirst  letter  stands  first,  and  its  other  letters  fol- 
low it  in  any  order? 

10.  If  every  permutation  of  two  or  more  letters  made  a 
word,  how  many  words  could  be  formed  from  10  letters? 

11.  In  how  many  permutations  of  n  letters  will  the  first 
letter  retain  its  place?  The  second  letters  retain  their  second 
places?     The  last  letter  retain  the  last  place? 

12.  If  we  write  under  each  other  all  ])ossiblo  ])ermutations 

m 


of  the  lirst  n  numbers  1,  "Z, 
of  each  column? 


n  what  will  be  the  su 
Ans.  l{n-\-l)\ 


AX 


PERMUTATTONS  AM)   COMBINATIONS. 


533 


Oils. 

cd  1,  2 a. 

p 

g  bti  (lono  ill 
s  were  to  bu 

in  single  filo, 
c  arningcd? 
nts  ill  the  lusL 
s  must  bo  to- 

Ictters  in  any 
tters  could  be 

formed  were 
)f  a  set  sliould 
ould  there  be, 

n  a  row.  In 
the  condition 

'oblem  in  case 

the  words  yon 
will  all  throe 
may  be  rocog, . 
her  letters  fol- 

etters  made  a 
.0  letters? 
will  the  first 
n  their  second 

])ermutations 
11  be  tlie  sum 


{n-\-l)\ 


13.  What  will  be  the  sum  of  each  cohiiun  if  the  ]i()ssiblo 
permutations  of  the  ligures  1  'l  'I  ;j  ;}  ;}  4  are  all  ^\^iUell  under 
each  other? 

14.  From  a  collection  of  .")  capital  letters  and  7  small  ones 
how  many  combinations  of  1  capital  witii  'I  small  ones  can  be 
formed? 

15.  The  driver  of  a  four-horse  coach  can  choose  his  horses 
from  a  stable  of  G  white  and  8  black  horses,  but  he  must  not 
pair  2  horses  of  dilTerent  colors.  In  how  many  dillerent  ways 
may  he  choose  his  -i  horses? 

16.  Uow  many  of  the  ])ossible  combinations  of  .'3  letters 
in  the  first  10  will  contain  the  letter  c'i  How  many  will  con- 
tain both  the  letters  c  and  r/? 

17.  Of  the  jiossiblu  combinations  of  s  things  in  u,  how 
many  will  contain  a  designated  thing?  How  many  2  desig- 
nated things?     How  many  k  designated  thir  .' ?: 

18.  A  party  of  G  meet  for  whist,  2  waiti»  't  W:  i  the  other 
4  play.  Each  4  must  jday  one  game  witl.  t>  ah  possible  ar- 
rangement of  partners.  How  many  games  ":d  be  played  in 
all;  how  many  will  each  person  play,  a'  d  how  many  times 
will  any  two  designated  persons  have  me^.   :^  partners? 

19.  From  a  collection  of  5  letters  and  G  numbers  how 
many  combinations,  each  consisting  of  1  letter  and  2  num- 
bers, can  be  formed?  How  many  consisting  of  2  letters  and 
3  numbers?     Of  5  letters  and  4  numbers? 

20.  From  a  collection  of  ;//  letters  and  n  numbers  how 
many  combinations  of  /•  letters  with  .v  numbers  can  be 
formed? 

21.  In  how  many  wjiys  may  a  pile  of  20  balls  be  divided 
into  two  piles,  the  one  having  15  balls  and  the  other  5? 

22.  How  many  dillerent  signals  may  be  made  with  4  fl.'igs 
of  dillerent  colors,  it  being  assumed  that  each  difl'erent  ord(a' 
of  each  combination  forms  a  dilTerent  signal,  ])ut  that  the 
siernal  remains  the  same  when  the  order  is  reversed? 

23.  What  would  be  the  answer  to  the  ])receding  ])ro])]em 
if  each  combination  of  several  flags  could  be  ai-ranged  either 
horizontally  or  vertically,  and  an  inversion  of  each  vertical 
arrangement  nuule  a  dlilereiit  signal? 


Il 


r)'M 


rKRMUTAriOS.H  AND   COMlUNATIONFi. 


24.  Ilosv  niuny  difforont  aipfiiiils  oiiii  be  niiide  with  10  fluc^a, 
of  wliicli  'Z  iU'O  white,  '<)  red,  mid  5  blue,  nil  lioisLed  together 
in  u  vertical  row? 

■2<^.  How  iiiiiny  different  arrangoments  can  bo  made  of  a 
base-ball  "  nine,  '  supposing  that  only  one  man  can  pitch,  and 
only  two  can  catch? 

26.  Supposing  tliat,  in  a  game  of  cliess,  the  first  player 
always  nas  a  choice  of  two  good  moves  and  the  second  player 
of  tiireo,  how  many  games  of  ^0  moves  each  are  possible? 

27.  If  the  8  i)ieces  at  chess  could  be  arranged  in  any  onkir 
on  the  8  sfpiares  of  the  first  rank,  how  many  different  arrange- 
ments would  be  possible? 

28.  In  how  many  different  ways  can  4  pawns  })o  arranged 
ni)on  the  G-t  squares  of  a  chess-board?  How  many  different 
ari'.ingernents  can  be  made  witii  a  king,  queen,  knight,  and 
rook?     Explain  the  relation  of  the  two  answers. 

29.  In  how  many  ways  may  \'i  balls  be  divided  into  throe 
piles,  containing,  the  one  3  balls,  tiie  second  4,  and  the  tliird5? 

30.  l\\  how  many  ways  may  n  balls  bo  divided  into  3  piles, 
containing,  the  one  /;,  the  second  7,  and  the  third  /•  balls 
{p -\- q  ^- r  =  n)^ 

31.  What  must  bo  the  value  of  r  in  order  that 

/-'"      _  n^     ^ 

32.  Tiio  ratio  of  the  number  of  combinations  of  2n  things 
in  4:11  to  that  of  Ihc  combinations  of  n  tilings  in  27i  is 

(2m  +  1)(2?^  +  3)  ....  (An  -3)(4n-  1) 
1.3.5  ....  (2  ?i—  1) 

33.  Show  that  the  sum  of  the  7i\  different  numbers  that 
c;in  be  formed  by  permuting  any  w  different  digits  is  divis- 
ible by  {n  —  i)  times  the  sum  of  the  digits,  and  that  tlie 
(juotient  is  111  .  .  .  . 

34.  If  we  define  a  magic  square  as  an  arrange- 
ment of  m"  numbers  in  a  square  such  tliat  the  sum 
of  every  line  and  every  column  is  equal  to  the  same 
quantity;  sh.ow  that  if  one  such  arrangement  is  ])os- 
sible  with  given  numbers,  then  (w!)'  are  possible. 


6     1     S 


i 


5 


.> 


2     9     4 


8Eiuii:.i. 


635 


,vith  lOfliiirs, 


See  murgiu  for  exiiini)le  of  sciiiiiic  wluii  n  —  H,  and  uolc  that  we 
leave  out  of  ouusideration  the  diaguiiul  liiicd  of  uutiibors. 

35.  Givoji  m  (litTcrent  loiters  uiul  n  dilTi'ient  numbers, 
find  tlio  ntiHil)or  of  ditlurent  jK'rniutiitions  i-ucli  containing'  /• 
letturs  and  .s'  nunihtTs. 

36.  (liven  n  nnequal  stvai^lit  lines;  how  many  non-identi- 
cal rectangular  |)arallelo[»ipeds  may  be  formed,  eaeii  of  wiiose 
edges  must  be  e({ual  to  some  one  of  these  hnes  in  the  two 
cases;  (I)  Wiien  tho  same  line  cannol  be  repeated  in  a 
figure  and  (2)  When  it  can  l)e  re])eated  without  restriction. 

37.  The  same  tiling  being  supposed  and  case  (1)  taken; 
how  many  dillerent  parallelopipedons  may  be  built  u])()n  the 
same  horizontal  i)lane  as  a  l)aae,  with  their  vertical  faces 
toward  the  four  points  of  the  comj)ass  ;  two  figures  being 
regarded  as  dilferent  when  they  cannot  be  brought  into  coin- 
cidence without  turning  them  art)und  or  over. 

38.  Given  vi— I  sets  containing  respectively  2r^  3re .  .  .  .  na 
different  things;  show  that  the  number  of  combinations  com- 
prising a  of  the  first  set,  2«  of  the  second,  etc.,  is      ,  ,/. 

Series. 

Indeterminate  Coefficients. 

Develop  the  following  exi)ressions  in  powers  of  x  by  tho 
method  of  indeterminate  coeOicients: 
1  +  nx 


I. 

3- 
5- 
7- 
9- 

1 1. 


\-x' 
1  -\-  mx 

1  -\-nx' 
a  {a  -f  x) 

"a'  4-  ^'' 
x'  +  a' 
X*  -\-  a 


«• 


X 


(1  -  c.r)  (1 


_fv 


C  / 


1 


11^  -\-  ax  -f-  X 


3' 


2. 

4. 
6. 

8. 

10. 
12. 


1_-M 
1  —  nx' 

X  -\-  a 

c  —  x' 

a  -1-  :/• " 


X 


[l  -:r)  {I  -  bx)' 


1  -\-x 


,»• 


a' 


\     

ax  -\-  X 


X 


030 


^•A7;//iX 


I  u 


I      '♦ 


fr  f 


PllO DUCTS  OF   SeUIEH. 

Form  I  lie  products: 

,.    (1  _  .,  .|.  .,'^_  ,;=•+  .  .  .  .)  (1  +  ,.  _^.  ^'_|_  .r'+  .  .  .  .). 

,.  (i  +  «^4.«v-f....)(l  +  ^  +  ^  +  ^+....). 

5.(l-«.  +  «V-....)(l-^  +  ^-^+....). 

6.  (1  H-^>x  +  .V  +  4.c'+  ....)'• 

7.  (1  -:2x'4-3a;'  -Ax'  -{■  .  .  .  .)\ 

Carry  tlio  products  as  far  as  .f*  and  express  the  n^^  term 
of  the  [)roduct  in  terms  of  n  m  cacli  case  for  which  you  can 
form  it. 

FiGURATE  Numbers. 

1.  Enumerate  an  incomplete  \n\Q.  of  cylindrical  shot  (§  288) 
having  n  shot  in  its  bottom  row,  and  as  many  in  its  top  row 
as  there  are  rows. 

Show  tliat  in  this  problem  the  number  n  must  be  odd. 

2.  The  toj)  and  bottom  rows  of  an  incomplete  pile  of  cylin- 
drical shot,  havinn^  8  rows  in  all,  contain  9  shot  less  than  one 
third  the  pile.     How  many  shot  are  in  the  i)ile? 

3.  Tn  an  incomj)leto  pile  of  03  cylindrical  shot  35  are  in 
the  interior  of  the  pile,  so  as  to  be  com})letely  surrounded  by 
others,  and  28  form  the  top,  bottom  and  sides.  Describe  the 
)>ile,  and  show  that  two  piles  may  be  formed  which  fuITil  the 
conditions. 

4.  Tn  a  triaiifi^ular  pyramid  of  balls  the  ratio  of  the  wliole 
number  of  balls  to  the  number  in  tlie  bottom  layer  is  14  :  3. 
How  many  balls  form  the  ])ile? 

5.  In  a  trianjjular  pyramid  having  n  balls  on  each  edge, 
how  many  balls    orm  the  four  faces? 

6.  If  20  balls  in  a  triangular  jiyramid  are  completely  sur- 
rounded by  others,  how  many  form  the  entire  pyramid? 


|V....). 


tlic  ?i'''  term 
liicli  you  can 


il  shot  (§288) 
ill  its  top  row 

st  be  odd. 
I  pile  of  eylin- 
less  tliiiii  one 

hot  35  are  in 

nrroiiiidcd  by 

l)c.seril)e  tlic 

licb  ful>il  the 

•  of  tlie  Avliole 
tiyer  is  14  :  3. 

n  each  edge, 

nipletely  sur- 
y  ram  id? 


.s/i7k7/;.S. 


r);a 


'     7.  A  rectangular  i)ilc  has  15  balls  in  its  top  row  utid  its 
lesser  side  has  lU  balls.     Ki'umerate  the  balls  iti  the  pile. 

8.  If  one  side  of  the  base  contains  m  balls  atid  the  other  ;/, 
{m  >  h),  how  many  balls  will  the  pile  contain;  how  many 
layers,  and  how  many  b;dls  in  the  toj)  row? 

9.  If  -195  l)alls  form  a  complete  rectangular  jule,  bavin*'' 
10  balls  on  one  side  of  the  base,  how  many  will  the  other  side 
comprise? 

10.  How  many  balls  in  a  square  i)yrami<l  Iniving  U  balls 
on  each  side  of  the  base? 

11.  A  rectangular  pile  has  84  shot  in  its  bottom  layer  and 
GO  in  the  next  layer,     ilow  many  in  the  whoh;  pile? 

Prove: 

12.  1.2  +  ^.3  +  3.4+  ....  4-  n{n-\-  1) 

13.  l7i  +  i>(M-  1) -f- 3  (?i- :>)+....  +  n[it-{n-  1)J 

=  ''  (''  +  ^ )  0'  +  -) 
3! 

14.  l.^-f  2.4  +  3.0-1-  .  .  .  .  -{-  n.2n 

;"3 

15.  1  (2  -  w)  +  5i  (4  -  n)  +  3  (0  ~  /O  +  •  •  .  .  +  n' 

_n{n-\-  i){n  +  '^) 
3! 

16.  If  we  multiply  the  corresponding  terms  of  the  two 
progressions: 

a,  a  -{-  hf     a  -f-  2A,  .  .  .  .  (f  -{-  iJi, 

b,  b  —  h,     b  —  2h,  .  .  .  .  b  —  ih, 

the  sum  of  the  products  will  be 


{i-\-l)  \ah  + 


ih  {b  -  ^0  _  i{2i-{-^)  Jr 
2  6 


17.   Find  the  sum  of  the  products  when,  in  [lie  second 
series,  the  C.  D.  is  +  h  instead  of  —  h. 


fl  ^ 


r)'i8 


SERIES. 


%  H 


\   .    ♦ 


Express  tlie  values  of 

icS.  ab  +  (a  +  h)  [b  +  k)  +  {a  +  2//)  {b  +  U)  +  .  .  .  . 

to  n  ternirf. 

19.  1  .:J  f  3 .5  +  5 . T  +  .  .  .  .  +  i  (/  4-  ^). 

20.  1«  +  3  {a  —  3)  -f-  5  (^e  —  G)  -f  •  •  •  ■  to  n  terms. 

21.  1  .r<  -f-  3  (^r  -f  3)  +  5  (rt  +  6)  -f-  .  .  .  .  to  ;i  terms. 

22.  Prove  the  eiiuutions: 

1.2.3  +  2.3.44-3.4.5  +  4.5.0  =  ^-— 


(1) + e) + (!)  =  (D- 


by  subtrjicting  from  the  second  member  tlie  successive  terms 
of  tlie  first  member,  beginning  Avitli  the  hist. 

23.  Generalize  the  preceding  result  by  ju-oving  in  the  same 
way  the  general  e(iuation: 


;) + (^-' 1 + 


• + 1-7 


}( 

S 

,l-^r  1 


Note  thiit  tlie  first  operation  will  be  to  deduce 

\s+  1/  ~  U/       V+  l)- 

By  means  of  the  preceding  formuljB  write,  on  sight,  the 
values  of: 

24.   1 .2.3.4  +  2.3.4.5  +  3.4.5. G  +  4.5.6.7 

^^'    1^2.3  "^1.2.3"^  lT2".3  "^  OV3  "^  073' 

26.  1.2.3.4 +  2.3.4.5  +  .  .  .  .  +  7/(7^  +  1)  (?i+2)  (;/+3). 

27.  Show  tiiat  the  sum  of  the  ])roducts  of  tlie  first  n  natu- 


ral numbers  taken  bv  2's  is 


(j^_Z:  ^ LZi (^+  1)  (3//.  +  2) 
2  • 


2<S.  In  the  following  scheme  we  start  with  a  column  of  /y's 
on  tlic  left,  and  with  the  top  line  a,  (3,  y,  S,  etc.  Then,  eacli 
number  foHowing,  in  each  column,  is  formed  by  adding  the 
Dumber  ai)Ove  it  to  the  luimber  on  the  left  of  the  latter.     It 


to  n  torriH. 

)  n  terms. 

^  n  terms. 

).6.7 
4 

;cessivc  terms 

gin 

the  same 

1)    1    M 

= 

1,1  + 

D- 

(7) 

■) 

on 

sight, 

the 

.3* 

e  first  ?i  natu- 
\n  4-  2) 

« 

cohimn  of  ^v's 
.  Then,  eacli 
hy  adding  tlie 
,he  latter.     Ifc 


SEUIKS. 


^>:v^ 


is  now  required  to  write  tiic  general  expression  for  llic  ii\ 
number  in  ihe  iid,  lid,  -iiii,  and  /lii  columns. 


y 

fi 

4/i  4-  6(t 


6  +  4y  +  G/i  -I-  4  a 


29.  A  trader  starts  with  a  cai)ital  of  a  dollin>;  lie  gains, 
and  adds  to  his  cajjital,  b  dollars  tjie  first  year,  and  r  dollars 
more  each  year  than  he  did  the  year  before.  Kxnicss  his 
accumulated  cai)ital  at  the  end  of  71  years  in  terms  of  n. 

SuMMATiox  or  .Seiues. 
Sum  to  infinity: 
I.   l  +  u  +  {i  +  2n)x+{\  +  Zn)x'j-{\    \   •\n)x'-{- 


2.  1  +3./-+  r..r'+  ]().,•'-{-  . 

3.  1  +  4.V  +  'Xc'  -f-  lilr'  -{- 
1.1.1 


+ 


n{h  -I-  1) 


^11  -  1 


I 


-\-7l'x"-'-\- 


+  :r-T  -^  rr-,  + 


1.3    '    2A 


2.5 


+  0  + 


0.0 


8.11 


+ 


^-  274  +  4:0  +  (iTs  + 


+ 


3.4    '    3.4.5    '    4.5.0 


+ 


-f 


8. 


.) ..) 


^- 


•.'.-!.(; 


1.4 


H- 


s 


-f 


<;.!) 


-i- 


+ 


,0.    ]+{a+\+a-')r+  {(/'-{-(!+  1  +>(-'-]  ,'-'),■'+ 
M.   {n+  I)V  ]    (;/.4-:>)V  +  (;/   f  3)V'|    .... 
1  •'  *?  4 


N 


n 


rA(> 


LIMITS. 


Sum  to  n  terms: 
13.  ce-{-{a^-iy^-{a-\-^iy-\- 


14.   2  +  5  +  9+  14  + 


-1- 


n{n-\-':\) 


15.  3  +  8  +  15  +  24  +  .  .  .  .  +  ;/.  {n  +  2). 

16.  1  +  ^•  +  2  (2  +  /t-)  +  3  (3  +  Ar)  +  .  .  .  .  +  vi  (;i  +  ^•). 
iGa,  Show  liiiit  the  scries: 

2^3     4^5     •  •  •  • 

may  be  tniusformod  into  eitiier  of  the  tlirec  forms: 

Jl     ,   _i   _lJ_     , 
1.2'^3.4"~^5.G^  •  •  •  • 

2.3       4.5       (5.7        •  •  •  • 

1  .  _1_  I  __L   .  ._J_  4.  _L_  , 

2  ^1.2. 3  ^3. 4. 5  ^5.0. 7^7. 8. 9^'  '  *  ' 

17.  How  do  two  of  the  i)recei.ling  results  enal)lo  us  to  sum 

T'i  '*"  2~:J  "^  3~4  "^  *  '  *  *  ^^'^  i^'fi^^itum? 

18.  What  nuuiber  is  equal  to  the  co.ntinucd  product: 

2i.4i..Si=c.K;a'j.32A  .  ...  ad  infinilnm? 

19.  To  \vhat  limit  ai)i)roachcs  the  indcriuitely  continued 

])ro(luct: 

'     ?_     'L     1_ 
«'^cr"'.^/'i'.a"'  .  .  .  .  ? 


or 


or 


Limits. 


Find  the  limits  of 


{.r  +  ffY 


as  X  increases  indelinitely. 


ax 

nx 


a    (< 


ii.     a 


i< 


(( 


tl 


it 


-f-  n  {n  +  ^•). 


ais: 


ible  us  to  sum 

urn? 

product: 

m? 

,ely  continued 


LIMITS. 


.041 


X  —  a 


4-  -j ^       its  rr  api)rouclics  c?  iudelinitely. 


(<  (( 


X       a 
x"  -  a' 
x'  -  (? 
^,       2; 


G.      — ''      '* 


<< 


(( 


(i 


t( 


{( 


(C 


7. 

8. 
9 


:f  —  « 

{x-\-(fy  -  (.r  -  ^/)« 


,:;i-r ^if^  -''  increases  imJuiinitely. 


(1    +    /A'f 

(I  —  a.r)" 


a    (( 


10. 


II. 


12. 


•    (1  -Ijj'^ 

V  +  ^'^  +  3'^  +  4^  +  . 


(<   (( 


(( 


« 


.  n" 


n' 


as  w  increases  indelinitely. 


r  -f-  2^  4-  3"  -f -^n' 


n' 


Jr-h -"'  H-  3"*  +  4"*  + 


+  «• 


/i'"+i 


(( 


(t 


13.  The  first  term  of  a  series  is  -,  the  second  ~  ?-,  and 

'*  0 

cacli  succeeding  term  oi.e  lialf  the  sum  of  the  two  whicli  i)rc- 
cede  it.  To  wliat  limits  will  the  nth  term  and  the  sum  of 
the  series  rpproach  as  n  increases  indefinitely? 

14.  Find  the  limit  toward  which  the  ni\\  term  approaches 
when 

First  term  =  a  -{■  2b;    second  term  =  a  —  b; 

1; 


it      i(    


a: 


each  term  after  the  second  being  half   the  sum  of  the  two 
preceding  terms. 

15.  The  first  term  of  a  series  is  a,  the  second  h,  and  each 
following  one  the  geometrical  mean  of  the  two  i)receding  it. 
Show  that,  as^iincreases  indefinitely,  the  ?ith  term  apiU'Oiiclies 
the  limit  a'ibl. 


.042 


niNOMlA  L    TUKOIUCM. 


Binomial  TlKorc^iu. 


I)c 

velop: 

I. 

(1  -  xY\ 

3- 

(1  -.r)-\ 

/           I  \'" 

5- 

\        ml 

7- 

('.'  4-  h)K 

9- 

{a  +  /.)-♦. 

1 1. 

.T+     J. 

/                   1    \'-'H 

2, 

(1-.')-^ 

4- 

(1           .'•)-". 

0. 

8. 

(./  -  />)i. 

lO. 

(^^  -  /v)-*. 

12. 

/          IV 

14. 

/                 1    ^^2U 

V'"  +  :J  • 

,6.  ^..--^-.j        . 


111  the  six  List  (li'velopineiits  Jirnmge  the  result  in  the  fdi'in 

Develop  as  far  as  x^  and  arrani^e  in  powers  of  x\ 

17.    (1  4-.r +  .t')".  18.   (I  -  .rf. .'•■;)". 

19.    (1  -f  X  —  .r')-".  20.    (1  —  X  -  -••■)-«. 

21.  Write  the  development  of  (1  —  .'r)~*  iii  siirlii  a  form 
thiit  I  he  deiiominator  of  eaeh  term  shwll  he  no  .-'  press  the 
/Hi  term  as  the  ])roduet  of  3  faetors. 

22.  Writ(^  the  developirient  '>f  (1  —  .r)"^"  in  .-ixh  a  form 
th.it  ;dl  I  he  tern;.-  jfliuli  httvc  die  common  di nominator 
{'\n  ~  1)1  and  show  that,  ])iiitin^^  f<  '■  brevity,  p  —  n  -|-  /  —  1, 
the  /til  term  may  be  written  in  tlie  foiin 


(/fi-  1)": 


,.i-l 


23.  Show  that,  if  71  be  an  odd  number,  and  if  we  j)iit 
p  =  71  -\-  2i  —  2,  the  ith  term  in  the  devclo])nicnt  of  (1  —  a)"" 
may  be  expressed  in  the  form 

(/  -  1')  (/  -  ^')  (p'-^n [/>'-  (n  -  2)'] 

2AJ'>  .  .  .  .  2(w  -  1) 


X 


<-i 


'W- 


J. 


nn\OMlAL  THEOREM. 


543 


24.  .Show  tliiit  the  ratio  of  tlic  nth  U-rni  to  tlie  {n  —  l)tli 
in  the  ex])rcssioii  of  (1  —  .r)~"  is  "Xx. 

Of  what  (juantities  are  the  following  scries  the  develop- 
jnents? 


27. 


1.3.5 


¥^ 


I   .  i.;j     1.3.5 


+ 


>■) 


-  -^ 


4.0 


1.3 


1  -h  T  ^^  +  f\.  /''  + 


1  +  7.-  + 


4.8 
1     .     1.3 


1.3.(1 

4.s,l: 


¥^ 


•      •      • 


+ 


1.3 


G.L-i  '  <;.iv>.is 


+ 


ill  the  fnrni 


Express  the  general  term  of  the  r(»ll()wing  developinents; 


29 
3' 

34 
35 

3^' 


,'\n 


3\|J 


(1  +  2.r  +  .r') 

(1  -  v>.M-./-') 

( L  4-  .r  4-  .r'  -f  .r'  -f  .  .  . 

(1  -.r +  .r'-a:'+  .   .   . 

(1  -  'lx^'l\e  -2V  + 


a  \  ~  n 


30.   (1  +;.'..  +  . r)- 

32.    (1  -•.'./•  +  . '••^ 

^/^/  iiijiiiiliony 

<<  a  \- 


rove 


that 


2"*  - 


m 


.)m  —  1 


+ 


f  (-  ir=i 


37.  If,  in  the  development  of  (1  +  r)"  we  call  the  second 
term  a  and  the  third  h,  express  n  and  x  in  terms  of  a  and  />. 

Of  what  expressions  are  the  following  series  the  develoi)- 
ments? 


39' 


41, 


m 


3*"+     .     3— '-f  (  J3— »  + 


ui 


+  1. 


(^) 


i)> 


3'""'+    -T    3'"-''- 


40.        1  4- 1  -I-  -  4- 


4^5    ,     4.5.G 


+ 


1  +  .'^  + 


n 


hi,,    («-f- !.)(«-    2) 


2>i 


.r'  + 


'^'// 


3^/ 


•+ 


42.  If  tr  1  "  the  7"ih  term  in  the  expansion  of  (1  -f  x) 


n  \r 


show  that 


^-f/2-h^-h 


=  (1  -  ^r 


(n  »  8) 


*--?pp« 


644 


LOGAJiJTJIAIS  AM)   LOUARITIIMIC  SlJIilliJS. 


Exponential    Tlu'oroni. 

1.  Find  two  exprcstiions  each  for  tlie  coeflicients  of  ./•',  .r\ 
and  ./■"  in  the  dcvelopnienr  of  e"c',  and  show  tJieir  identity. 

2.  Develo})  e""  in  ])owers  of  x  to  six  termy. 

3.  What  is  the  coeHicient  of  x"*//^  in  the  development  of 
e^+«?    In  tliat  of  c'-"? 

4.  Multiply  the  two  developments: 

,S                     3 
C-  :^    1    +  ;r  -f      •',,  +  ;'-  + 


1  .: 


•  ) ; 
,3 


and  show  by  what  relations  among  the  coefficients  the  prod- 
uct reduces  identically  to  unity. 

5.  Show  by  what  relations  the  development  of  c^-^  becomes 
identical  with  the  s(|uare  of  that  of  e". 

IjojJTiiritlinis  and    Lojj^aritliniic  Series. 

I.  Express  the  logarithm  of  the  continued  product  of  all 
the  terjiis  of  a  geometrical  progression. 

Calling  b  the  arbitrary  base  of  the  system  of  logarithms, 
solve  the  following  equations  so  as  to  express  x  in  terms  of  //: 


2.  log  X    =  y, 

4.  log  'Ix  —  y. 

6.  log  nx  =  my. 

8.  log  .r"  —  my. 

10.  y  —  r*'"*^' 


3.  log  ;r     -  ay. 

5.  log  mx  =  a  -\-  y. 

7.  log  .r'     =  y. 

9.  ,/  =  Z;'°«f^ 


-logj 


II.  ma'"*'-'   =  y 
Ueduce  to  their  simplest  form  the  exj)ression8: 


a 


12. 


lope 


»3- 


Ort^HIO..    (;1.>K'^. 


Pr 

ove 

the 

ideii 

tities: 

r  1. 

>(r  V 

(I 

(I 

14. 

7)1 

".r"  = 

=  /;"' 

ogmx 

• 

»5- 

'h' 

"«»« 

.^.loK 

"'«/' 

'^8n. 

^v/i^s. 


3nts  of  .r\  .i'\ 
ir  iilentitv. 

ivclopmcnt  of 


311  ts  tlie  prod- 


of  e^^  becomes 


Series. 


product  of  all 


)f  logarithms, 
in  terms  of  //: 

:  n?/. 
--  a  -\-  y. 

'-  !/' 
--  //"«*. 

=  ?/'• 

ns: 


,loK  n 


LOGAUrniMii  ASD   LOGAlilTJIMlC  i^tJJUh'S.         ;-)4.") 

i6.   If  a,  h,  and  c  be  the  mth,  pt\\  and  qih  terms  of  a  geo- 
metrical ])i'0{,n-essioii  show  that 

{p  —  q)  log  a  -f  (r/  —  yu)  log  A  -f-  (///  —  p)  log  6-  =  0. 

17.  Prove  that,  the  value  of  the  exprej^sion 

is  independent  of  n  and  e<iual  to  h)g  n. 

18.  Prove  tiie  eiiuation: 

53  log  X  —  h)g  {x  -f  a)  -    h)g  (,r  —  n) 


I  \  ^ 


( 


+  1 


a 


.0  -f 


I 


^/ 


1.:.  + 


19.  If  <t,  b  and  c  are  tliree  conseciiiivc  iiiiihIkts  A\o\\'  tliat 
2  l„gA  -  l,>g«  -  log  r  =  ■IH\  ^-^-  +  .|-,5j.  4.-1^5  +  •■••[• 

20.  Prove: 


o 


Nap.  log  4  -  1  +  ^--  -f  3_^-_-5  +  ^-^^  +  .  .  .  . 

21.  If  ^/,  /y,  r,  ^i,  etc.,  are  in  geometrical  progresaion,  then, 
in  order  the  equations 

I         I        »         I 

am    =:  b"  =  CP  =  (I'l  —    .    .    .    . 

may  be  satisfied,  the  quantities  in,  ?^  p,  q,  etc.,  must  be  in 
arithmetical  progression. 

22.  If    //    =    ]0i-»"<?^     and     2  =  10»-'"K<',     show    that 
^1 

X  =:  lor-'oK^ 

23.  Prove  the  development 

log  (I  -2.r  +  .r')  =  -2(.r4-  lar'-f  .y;r'-f-_^.r'4-  .  .  .  .) 

and  by  making  the  development  m  another  f(<rm  and  com- 
paring the  coeMlcients  of  .r"  prove  the  identity 


n 


n 


-3 


>»  — *  J- 


(«-4)(« -r.).,„_ 


)  II — fi 


1.2.3 


the  sj'ries  terminating  with   the  last  exponent  which  is  not 
negative. 


IT. 


i  t 


HINTS  UN  A  COURSE  IN  Ab\  ANCi;i)  AL(;i;i',RA. 


For  the  Ix'nt'fit  of  students  who  nmy  n)ut('!iii)liit«'  ii  nmrsc  of  rcniiiii;; 
in  tlic  various  Ijianclics  of  Advanccil  Alj^cltrii.  tin-  tollowin;,'  list  of  miI)- 
j«'ctH  and  books  Iiuh  l)ct'H  prepared.  As  a  ^MMnral  rule,  the  most  eKteuded 
mid  tlioroujfli  treatises  are  in  tlie  (Jennau  Lan^nuiije,  while  the  Kreuch 
works  arc  noted  for  cleganco  and  simplicity  in  treatment. 

To  pursue  an,v  of  these  suhjects  to  advauta^'e,  tiie  stu(h'nt  nhould  lie 
familiar  with  tho  l>illi'ren'ial  Calculus. 

I.  TllK    (JENMWAJ.   TIIF.ORY  OF    P:(2UATI0NS.— In   Enfrli.-h.  ToD- 

iiUNiKii's  is  tho  work  most  road. 
Seruet,  A/'/rhir  Si/prnn/ir,  2  vols.,  8vo,  is  the  stnnda'd  French  work, 

covering  all  thi^  collatt'ral  sid)jects, 
Jordan,  Tluuric  dcs  Subditufitiiis  ct  dtx  Aqimti'iun  A/f/i'hiif/ii<fi,  1  vol.,  tto. 

is  tho  hu-j^c^st  and  most  exhaustive  treatise,  but  is  too  abhtruae  for 

nny  but  experts. 

II.  DETEllMIXANTS— Hai.tzeh.  T/ienrir  ih r  Ihtrnuiinnitm,   is  the 

standard  treatise.  There  is  a  French  but  no  F.u^lish  translation. 
A  recent  Knglish  work  is  Hoiti;i{T  F.  Scott,  T/ic  'f/fon/  of  Ihtvr- 
minaiits  <tiid  their  ApplicdtioitH  in  Antdi/nia  it  ml  divmetrf/. 

III.  THE  MODERN  IIKJIIEU  ALOEBHA,   resting  o)i  the  the(.ri.".s  of 

Invariants  'ind  Covariants. 
Salmon,   Lr.swns  iidrDdarfori/  to  (he  Modern    IIi'jh(r   A^fjiftni,  is  the 

standard  English  work,  and  is  espirially  adapted  for  instruciion. 
Clebsch,  Theorie  der  him'iren  AUjeliraiHrhen  Fornim,  is  more  exhaustive 

in  its  special  branch  and  recjuires  more  lamiliarity  with  advanced 

systems  of  notation. 

IV.  TIIE  THEORY  OF  NUMBERS.      There   is   no   recent  tiratise   in 

En<,dish.  Gauss,  Dinquidtiones  Arithmctinp,  and  liKOKNOUK, 
Theorie  den  Nomhres,  an^  the  old  standards,  but  the  latter  is  ran; 
and  costly.  Li-:,iki:nk  Di!{ICIII,f:t,  Vorlennngen  iiher  ZahJeidheorie, 
is  a  good  (Jerman  Work.  There  is  also  a  chaiiter  on  tht-  subject  in 
SKitiiET,  Alf/rbre  Snpericnrc. 

V.  SERIES.— This  subject  belongs  for  the  most  i)art  to  the  Calculus,  but 

Catalan,  Trade  elhncnUdre  des  S'rii.s,  is  a  very  convenient  little 
French  work  ou  those  Series  which  can  be  treated  by  Elementary 
Algebra. 

VI.  QUATERNIONS.— Tait,   Elemrnfari/    Treatise  on  Quaternions,  is 

prepared  especially  for  students,  and  contains  many  exercises.  The 
original  works  of  ]\\}>iu.TO'S,  Lectures  on  Quaternions  and  Elemcnta 
of  Quaternions,  are  more  extended,  and  the  latter  will  be  found 
valuable  for  both  reading  and  reference. 


